Theorem onprcf1of1ord 36055

Description: If 𝐹 is a bijection from the ordinals to the universe, 𝐻 is a bijection from the ordinals to 𝐴 if 𝐴 is a proper class, and 𝐼 is a bijection from the ordinals to 𝐵 if 𝐵 is a proper class, then 𝐽 either maps 𝐴 one-to-one into 𝐵 or maps 𝐵 one-to-one into 𝐴. This is the ZFC version of (5 → 10) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the first three hypotheses would be replaced by something like (𝜑 → ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:On⟶1-1onto𝑋)). However, we cannot quantify over classes, so we split this into the case where 𝑋 is the universe, the case where 𝑋 is an arbitrary proper class 𝐴, and the case where 𝑋 is an arbitrary proper class 𝐵, which together are sufficient for this proof. (Contributed by BTernaryTau, 17-Sep-2026.)

Hypotheses

Ref	Expression
onprcf1of1ord.1	⊢ (𝜑 → 𝐹:On⟶1-1ontoV)
onprcf1of1ord.2	⊢ (𝜑 → (¬ 𝐴 ∈ V → 𝐻:On⟶1-1onto𝐴))
onprcf1of1ord.3	⊢ (𝜑 → (¬ 𝐵 ∈ V → 𝐼:On⟶1-1onto𝐵))
onprcf1of1ord.4	⊢ 𝐽 = if(𝐴 ∈ V, if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))), (𝐻 ∘ (◡𝐹 ↾ 𝐵)))
onprcf1of1ord.5	⊢ 𝐾 = (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)})

Assertion

Ref	Expression
onprcf1of1ord	⊢ (𝜑 → (𝐽:𝐴⟶1-1𝐵 ∨ 𝐽:𝐵⟶1-1𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Allowed substitution hints:   𝜑(𝑥)   𝐻(𝑥)   𝐼(𝑥)   𝐽(𝑥)   𝐾(𝑥)

Proof of Theorem onprcf1of1ord

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onprcf1of1ord.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:On⟶1-1ontoV)
2	1	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → 𝐹:On⟶1-1ontoV)
3		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
4		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
5	2, 3, 4	onprcf1of1orbd 36054	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → (∃𝑓 𝑓:𝐴⟶1-1𝐵 ∨ ∃𝑓 𝑓:𝐵⟶1-1𝐴))
6		19.43 1917	. . . . . . 7 ⊢ (∃𝑓(𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴) ↔ (∃𝑓 𝑓:𝐴⟶1-1𝐵 ∨ ∃𝑓 𝑓:𝐵⟶1-1𝐴))
7	5, 6	sylibr 237	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → ∃𝑓(𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴))
8		dff1o5 6856	. . . . . . . . . . . 12 ⊢ (𝐹:On⟶1-1ontoV ↔ (𝐹:On⟶1-1V ∧ ran 𝐹 = V))
9	8	simprbi 503	. . . . . . . . . . 11 ⊢ (𝐹:On⟶1-1ontoV → ran 𝐹 = V)
10	1, 9	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝐹 = V)
11	10	rexeqdv 3324	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑓 ∈ ran 𝐹(𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴) ↔ ∃𝑓 ∈ V (𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴)))
12		rexv 3481	. . . . . . . . 9 ⊢ (∃𝑓 ∈ V (𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴) ↔ ∃𝑓(𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴))
13	11, 12	bitrdi 290	. . . . . . . 8 ⊢ (𝜑 → (∃𝑓 ∈ ran 𝐹(𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴) ↔ ∃𝑓(𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴)))
14		f1ofn 6847	. . . . . . . . 9 ⊢ (𝐹:On⟶1-1ontoV → 𝐹 Fn On)
15		f1eq1 6795	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹‘𝑥) → (𝑓:𝐴⟶1-1𝐵 ↔ (𝐹‘𝑥):𝐴⟶1-1𝐵))
16		f1eq1 6795	. . . . . . . . . . 11 ⊢ (𝑓 = (𝐹‘𝑥) → (𝑓:𝐵⟶1-1𝐴 ↔ (𝐹‘𝑥):𝐵⟶1-1𝐴))
17	15, 16	orbi12d 932	. . . . . . . . . 10 ⊢ (𝑓 = (𝐹‘𝑥) → ((𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴) ↔ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)))
18	17	rexrn 7111	. . . . . . . . 9 ⊢ (𝐹 Fn On → (∃𝑓 ∈ ran 𝐹(𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴) ↔ ∃𝑥 ∈ On ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)))
19	1, 14, 18	3syl 19	. . . . . . . 8 ⊢ (𝜑 → (∃𝑓 ∈ ran 𝐹(𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴) ↔ ∃𝑥 ∈ On ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)))
20	13, 19	bitr3d 284	. . . . . . 7 ⊢ (𝜑 → (∃𝑓(𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴) ↔ ∃𝑥 ∈ On ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)))
21	20	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → (∃𝑓(𝑓:𝐴⟶1-1𝐵 ∨ 𝑓:𝐵⟶1-1𝐴) ↔ ∃𝑥 ∈ On ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)))
22	7, 21	mpbid 235	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → ∃𝑥 ∈ On ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴))
23		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
24		nfrab1 3435	. . . . . . . . . 10 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}
25	24	nfint 4920	. . . . . . . . 9 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}
26	23, 25	nffv 6918	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)})
27		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
28		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
29	26, 27, 28	nff1 6798	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐴⟶1-1𝐵
30	26, 28, 27	nff1 6798	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐵⟶1-1𝐴
31	29, 30	nfor 1939	. . . . . 6 ⊢ Ⅎ𝑥((𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐴⟶1-1𝐵 ∨ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐵⟶1-1𝐴)
32		fveq2 6908	. . . . . . . 8 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)} → (𝐹‘𝑥) = (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}))
33		f1eq1 6795	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}) → ((𝐹‘𝑥):𝐴⟶1-1𝐵 ↔ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐴⟶1-1𝐵))
34	32, 33	syl 18	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)} → ((𝐹‘𝑥):𝐴⟶1-1𝐵 ↔ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐴⟶1-1𝐵))
35		f1eq1 6795	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}) → ((𝐹‘𝑥):𝐵⟶1-1𝐴 ↔ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐵⟶1-1𝐴))
36	32, 35	syl 18	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)} → ((𝐹‘𝑥):𝐵⟶1-1𝐴 ↔ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐵⟶1-1𝐴))
37	34, 36	orbi12d 932	. . . . . 6 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)} → (((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴) ↔ ((𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐴⟶1-1𝐵 ∨ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐵⟶1-1𝐴)))
38	31, 37	onminsb 7832	. . . . 5 ⊢ (∃𝑥 ∈ On ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴) → ((𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐴⟶1-1𝐵 ∨ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐵⟶1-1𝐴))
39	22, 38	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → ((𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐴⟶1-1𝐵 ∨ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐵⟶1-1𝐴))
40		onprcf1of1ord.4	. . . . . . . . . 10 ⊢ 𝐽 = if(𝐴 ∈ V, if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))), (𝐻 ∘ (◡𝐹 ↾ 𝐵)))
41		iftrue 4491	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))), (𝐻 ∘ (◡𝐹 ↾ 𝐵))) = if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))))
42	40, 41	eqtrid 2810	. . . . . . . . 9 ⊢ (𝐴 ∈ V → 𝐽 = if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))))
43	42	ad2antlr 740	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → 𝐽 = if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))))
44		iftrue 4491	. . . . . . . . 9 ⊢ (𝐵 ∈ V → if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))) = 𝐾)
45	44	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))) = 𝐾)
46	43, 45	eqtrd 2798	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → 𝐽 = 𝐾)
47		onprcf1of1ord.5	. . . . . . 7 ⊢ 𝐾 = (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)})
48	46, 47	eqtrdi 2814	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → 𝐽 = (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}))
49		f1eq1 6795	. . . . . 6 ⊢ (𝐽 = (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}) → (𝐽:𝐴⟶1-1𝐵 ↔ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐴⟶1-1𝐵))
50	48, 49	syl 18	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → (𝐽:𝐴⟶1-1𝐵 ↔ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐴⟶1-1𝐵))
51		f1eq1 6795	. . . . . 6 ⊢ (𝐽 = (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}) → (𝐽:𝐵⟶1-1𝐴 ↔ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐵⟶1-1𝐴))
52	48, 51	syl 18	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → (𝐽:𝐵⟶1-1𝐴 ↔ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐵⟶1-1𝐴))
53	50, 52	orbi12d 932	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → ((𝐽:𝐴⟶1-1𝐵 ∨ 𝐽:𝐵⟶1-1𝐴) ↔ ((𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐴⟶1-1𝐵 ∨ (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)}):𝐵⟶1-1𝐴)))
54	39, 53	mpbird 260	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ 𝐵 ∈ V) → (𝐽:𝐴⟶1-1𝐵 ∨ 𝐽:𝐵⟶1-1𝐴))
55	1	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ ¬ 𝐵 ∈ V) → 𝐹:On⟶1-1ontoV)
56		onprcf1of1ord.3	. . . . . . 7 ⊢ (𝜑 → (¬ 𝐵 ∈ V → 𝐼:On⟶1-1onto𝐵))
57	56	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ ¬ 𝐵 ∈ V) → (¬ 𝐵 ∈ V → 𝐼:On⟶1-1onto𝐵))
58		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ ¬ 𝐵 ∈ V) → ¬ 𝐵 ∈ V)
59		eqid 2763	. . . . . 6 ⊢ (𝐼 ∘ (◡𝐹 ↾ 𝐴)) = (𝐼 ∘ (◡𝐹 ↾ 𝐴))
60	55, 57, 58, 59	onprcf1of1orad 36053	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ ¬ 𝐵 ∈ V) → (𝐼 ∘ (◡𝐹 ↾ 𝐴)):𝐴⟶1-1𝐵)
61	42	ad2antlr 740	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ ¬ 𝐵 ∈ V) → 𝐽 = if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))))
62		iffalse 4494	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))) = (𝐼 ∘ (◡𝐹 ↾ 𝐴)))
63	62	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ ¬ 𝐵 ∈ V) → if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))) = (𝐼 ∘ (◡𝐹 ↾ 𝐴)))
64	61, 63	eqtrd 2798	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ ¬ 𝐵 ∈ V) → 𝐽 = (𝐼 ∘ (◡𝐹 ↾ 𝐴)))
65		f1eq1 6795	. . . . . 6 ⊢ (𝐽 = (𝐼 ∘ (◡𝐹 ↾ 𝐴)) → (𝐽:𝐴⟶1-1𝐵 ↔ (𝐼 ∘ (◡𝐹 ↾ 𝐴)):𝐴⟶1-1𝐵))
66	64, 65	syl 18	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ ¬ 𝐵 ∈ V) → (𝐽:𝐴⟶1-1𝐵 ↔ (𝐼 ∘ (◡𝐹 ↾ 𝐴)):𝐴⟶1-1𝐵))
67	60, 66	mpbird 260	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ ¬ 𝐵 ∈ V) → 𝐽:𝐴⟶1-1𝐵)
68	67	orcd 887	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ V) ∧ ¬ 𝐵 ∈ V) → (𝐽:𝐴⟶1-1𝐵 ∨ 𝐽:𝐵⟶1-1𝐴))
69	54, 68	pm2.61dan 825	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (𝐽:𝐴⟶1-1𝐵 ∨ 𝐽:𝐵⟶1-1𝐴))
70	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ V) → 𝐹:On⟶1-1ontoV)
71		onprcf1of1ord.2	. . . . . 6 ⊢ (𝜑 → (¬ 𝐴 ∈ V → 𝐻:On⟶1-1onto𝐴))
72	71	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ V) → (¬ 𝐴 ∈ V → 𝐻:On⟶1-1onto𝐴))
73		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 ∈ V)
74		eqid 2763	. . . . 5 ⊢ (𝐻 ∘ (◡𝐹 ↾ 𝐵)) = (𝐻 ∘ (◡𝐹 ↾ 𝐵))
75	70, 72, 73, 74	onprcf1of1orad 36053	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ V) → (𝐻 ∘ (◡𝐹 ↾ 𝐵)):𝐵⟶1-1𝐴)
76		iffalse 4494	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))), (𝐻 ∘ (◡𝐹 ↾ 𝐵))) = (𝐻 ∘ (◡𝐹 ↾ 𝐵)))
77	40, 76	eqtrid 2810	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → 𝐽 = (𝐻 ∘ (◡𝐹 ↾ 𝐵)))
78		f1eq1 6795	. . . . . 6 ⊢ (𝐽 = (𝐻 ∘ (◡𝐹 ↾ 𝐵)) → (𝐽:𝐵⟶1-1𝐴 ↔ (𝐻 ∘ (◡𝐹 ↾ 𝐵)):𝐵⟶1-1𝐴))
79	77, 78	syl 18	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝐽:𝐵⟶1-1𝐴 ↔ (𝐻 ∘ (◡𝐹 ↾ 𝐵)):𝐵⟶1-1𝐴))
80	79	adantl 487	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ V) → (𝐽:𝐵⟶1-1𝐴 ↔ (𝐻 ∘ (◡𝐹 ↾ 𝐵)):𝐵⟶1-1𝐴))
81	75, 80	mpbird 260	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ V) → 𝐽:𝐵⟶1-1𝐴)
82	81	olcd 888	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ V) → (𝐽:𝐴⟶1-1𝐵 ∨ 𝐽:𝐵⟶1-1𝐴))
83	69, 82	pm2.61dan 825	1 ⊢ (𝜑 → (𝐽:𝐴⟶1-1𝐵 ∨ 𝐽:𝐵⟶1-1𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090  {crab 3416  Vcvv 3454  ifcif 4485  ∩ cint 4910  ^◡ccnv 5654  ran crn 5656   ↾ cres 5657   ∘ ccom 5659  Oncon0 6380   Fn wfn 6553  ⟶1-1wf1 6555  ⟶1-1ontowf1o 6557  ‘cfv 6558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-oi 9531  df-har 9578  df-card 10047  df-ac 10222

This theorem is used by: (None)