Theorem fpwwe 10557

Description: Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 9058 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset ⟨𝑋, (𝑊‘𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 9940. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)

Hypotheses

Ref	Expression
fpwwe.1	⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}
fpwwe.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fpwwe.3	⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹‘𝑥) ∈ 𝐴)
fpwwe.4	⊢ 𝑋 = ∪ dom 𝑊

Assertion

Ref	Expression
fpwwe	⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹‘𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))

Distinct variable groups:   𝑥,𝑟,𝐴   𝑦,𝑟,𝐹,𝑥   𝜑,𝑟,𝑥,𝑦   𝑅,𝑟,𝑥,𝑦   𝑋,𝑟,𝑥,𝑦   𝑌,𝑟,𝑥,𝑦   𝑊,𝑟,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝑉(𝑥,𝑦,𝑟)

Proof of Theorem fpwwe

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7361	. . . . . 6 ⊢ (𝑌(𝐹 ∘ 1st )𝑅) = ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩)
2		fo1st 7953	. . . . . . . 8 ⊢ 1st :V–onto→V
3		fofn 6748	. . . . . . . 8 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ 1st Fn V
5		opex 5412	. . . . . . 7 ⊢ ⟨𝑌, 𝑅⟩ ∈ V
6		fvco2 6931	. . . . . . 7 ⊢ ((1st Fn V ∧ ⟨𝑌, 𝑅⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)))
7	4, 5, 6	mp2an 692	. . . . . 6 ⊢ ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩))
8	1, 7	eqtri 2759	. . . . 5 ⊢ (𝑌(𝐹 ∘ 1st )𝑅) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩))
9		fpwwe.1	. . . . . . . 8 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}
10	9	bropaex12 5715	. . . . . . 7 ⊢ (𝑌𝑊𝑅 → (𝑌 ∈ V ∧ 𝑅 ∈ V))
11		op1stg 7945	. . . . . . 7 ⊢ ((𝑌 ∈ V ∧ 𝑅 ∈ V) → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
12	10, 11	syl 17	. . . . . 6 ⊢ (𝑌𝑊𝑅 → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
13	12	fveq2d 6838	. . . . 5 ⊢ (𝑌𝑊𝑅 → (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)) = (𝐹‘𝑌))
14	8, 13	eqtrid 2783	. . . 4 ⊢ (𝑌𝑊𝑅 → (𝑌(𝐹 ∘ 1st )𝑅) = (𝐹‘𝑌))
15	14	eleq1d 2821	. . 3 ⊢ (𝑌𝑊𝑅 → ((𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌 ↔ (𝐹‘𝑌) ∈ 𝑌))
16	15	pm5.32i 574	. 2 ⊢ ((𝑌𝑊𝑅 ∧ (𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌) ↔ (𝑌𝑊𝑅 ∧ (𝐹‘𝑌) ∈ 𝑌))
17		vex 3444	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
18	17	cnvex 7867	. . . . . . . . . 10 ⊢ ◡𝑟 ∈ V
19	18	imaex 7856	. . . . . . . . 9 ⊢ (◡𝑟 “ {𝑦}) ∈ V
20		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
21	17	inex1 5262	. . . . . . . . . . . 12 ⊢ (𝑟 ∩ (𝑢 × 𝑢)) ∈ V
22	20, 21	opco1i 8067	. . . . . . . . . . 11 ⊢ (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹‘𝑢)
23		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑢 = (◡𝑟 “ {𝑦}) → (𝐹‘𝑢) = (𝐹‘(◡𝑟 “ {𝑦})))
24	22, 23	eqtrid 2783	. . . . . . . . . 10 ⊢ (𝑢 = (◡𝑟 “ {𝑦}) → (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹‘(◡𝑟 “ {𝑦})))
25	24	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑢 = (◡𝑟 “ {𝑦}) → ((𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))
26	19, 25	sbcie 3782	. . . . . . . 8 ⊢ ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦)
27	26	ralbii 3082	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦)
28	27	anbi2i 623	. . . . . 6 ⊢ ((𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦) ↔ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))
29	28	anbi2i 623	. . . . 5 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦)))
30	29	opabbii 5165	. . . 4 ⊢ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}
31	9, 30	eqtr4i 2762	. . 3 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
32		fpwwe.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
33		vex 3444	. . . . 5 ⊢ 𝑥 ∈ V
34	33, 17	opco1i 8067	. . . 4 ⊢ (𝑥(𝐹 ∘ 1st )𝑟) = (𝐹‘𝑥)
35		simp1 1136	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ⊆ 𝐴)
36		velpw 4559	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
37	35, 36	sylibr 234	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
38		19.8a 2188	. . . . . . . 8 ⊢ (𝑟 We 𝑥 → ∃𝑟 𝑟 We 𝑥)
39	38	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → ∃𝑟 𝑟 We 𝑥)
40		ween 9945	. . . . . . 7 ⊢ (𝑥 ∈ dom card ↔ ∃𝑟 𝑟 We 𝑥)
41	39, 40	sylibr 234	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ dom card)
42	37, 41	elind 4152	. . . . 5 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ (𝒫 𝐴 ∩ dom card))
43		fpwwe.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹‘𝑥) ∈ 𝐴)
44	42, 43	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹‘𝑥) ∈ 𝐴)
45	34, 44	eqeltrid 2840	. . 3 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥(𝐹 ∘ 1st )𝑟) ∈ 𝐴)
46		fpwwe.4	. . 3 ⊢ 𝑋 = ∪ dom 𝑊
47	31, 32, 45, 46	fpwwe2 10554	. 2 ⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))
48	16, 47	bitr3id 285	1 ⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹‘𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  Vcvv 3440  [wsbc 3740   ∩ cin 3900   ⊆ wss 3901  𝒫 cpw 4554  {csn 4580  ⟨cop 4586  ∪ cuni 4863   class class class wbr 5098  {copab 5160   We wwe 5576   × cxp 5622  ^◡ccnv 5623  dom cdm 5624   “ cima 5627   ∘ ccom 5628   Fn wfn 6487  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  cardccrd 9847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-en 8884  df-oi 9415  df-card 9851

This theorem is referenced by:  canth4  10558  canthnumlem  10559  canthp1lem2  10564