Theorem onprcf1of1orad 35965

Description: If 𝐹 is a bijection from the ordinals to the universe and 𝐻 is a bijection from the ordinals to the proper class 𝐵, then 𝐼 maps 𝐴 one-to-one into 𝐵. This is the ZFC version of (5 → 10) in https://tinyurl.com/hamkins-gblac in the case where one of the classes is a proper class. See onprcf1of1orbd 35966 for the case where neither class is a proper class, and see onprcf1of1ord 35967 for a theorem that covers both cases. Note that in NBG set theory the first two hypotheses would be replaced by something like (𝜑 → ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:On–1-1-onto→𝑋)). However, we cannot quantify over classes, so we split this into the case where 𝑋 is the universe 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
onprcf1of1orad.1	⊢ (𝜑 → 𝐹:On–1-1-onto→V)
onprcf1of1orad.2	⊢ (𝜑 → (¬ 𝐵 ∈ V → 𝐻:On–1-1-onto→𝐵))
onprcf1of1orad.3	⊢ (𝜑 → ¬ 𝐵 ∈ V)
onprcf1of1orad.4	⊢ 𝐼 = (𝐻 ∘ (◡𝐹 ↾ 𝐴))

Assertion

Ref	Expression
onprcf1of1orad	⊢ (𝜑 → 𝐼:𝐴–1-1→𝐵)

Proof of Theorem onprcf1of1orad

Step	Hyp	Ref	Expression
1		onprcf1of1orad.3	. . . . 5 ⊢ (𝜑 → ¬ 𝐵 ∈ V)
2		onprcf1of1orad.2	. . . . 5 ⊢ (𝜑 → (¬ 𝐵 ∈ V → 𝐻:On–1-1-onto→𝐵))
3	1, 2	mpd 16	. . . 4 ⊢ (𝜑 → 𝐻:On–1-1-onto→𝐵)
4		f1of1 6823	. . . 4 ⊢ (𝐻:On–1-1-onto→𝐵 → 𝐻:On–1-1→𝐵)
5	3, 4	syl 18	. . 3 ⊢ (𝜑 → 𝐻:On–1-1→𝐵)
6		onprcf1of1orad.1	. . . 4 ⊢ (𝜑 → 𝐹:On–1-1-onto→V)
7		f1ocnv 6837	. . . 4 ⊢ (𝐹:On–1-1-onto→V → ◡𝐹:V–1-1-onto→On)
8		f1of1 6823	. . . . 5 ⊢ (◡𝐹:V–1-1-onto→On → ◡𝐹:V–1-1→On)
9		ssv 3932	. . . . 5 ⊢ 𝐴 ⊆ V
10		f1ssres 6787	. . . . 5 ⊢ ((◡𝐹:V–1-1→On ∧ 𝐴 ⊆ V) → (◡𝐹 ↾ 𝐴):𝐴–1-1→On)
11	8, 9, 10	sylancl 598	. . . 4 ⊢ (◡𝐹:V–1-1-onto→On → (◡𝐹 ↾ 𝐴):𝐴–1-1→On)
12	6, 7, 11	3syl 19	. . 3 ⊢ (𝜑 → (◡𝐹 ↾ 𝐴):𝐴–1-1→On)
13		f1co 6791	. . 3 ⊢ ((𝐻:On–1-1→𝐵 ∧ (◡𝐹 ↾ 𝐴):𝐴–1-1→On) → (𝐻 ∘ (◡𝐹 ↾ 𝐴)):𝐴–1-1→𝐵)
14	5, 12, 13	syl2anc 596	. 2 ⊢ (𝜑 → (𝐻 ∘ (◡𝐹 ↾ 𝐴)):𝐴–1-1→𝐵)
15		onprcf1of1orad.4	. . 3 ⊢ 𝐼 = (𝐻 ∘ (◡𝐹 ↾ 𝐴))
16		f1eq1 6773	. . 3 ⊢ (𝐼 = (𝐻 ∘ (◡𝐹 ↾ 𝐴)) → (𝐼:𝐴–1-1→𝐵 ↔ (𝐻 ∘ (◡𝐹 ↾ 𝐴)):𝐴–1-1→𝐵))
17	15, 16	ax-mp 5	. 2 ⊢ (𝐼:𝐴–1-1→𝐵 ↔ (𝐻 ∘ (◡𝐹 ↾ 𝐴)):𝐴–1-1→𝐵)
18	14, 17	sylibr 237	1 ⊢ (𝜑 → 𝐼:𝐴–1-1→𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895  ^◡ccnv 5649   ↾ cres 5652   ∘ ccom 5654  Oncon0 6362  –1-1→wf1 6535  –1-1-onto→wf1o 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545

This theorem is used by:  onprcf1of1ord  35967