Theorem fpwwe 10684

Description: Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 9169 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 10068. 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 7434	. . . . . 6 ⊢ (𝑌(𝐹 ∘ 1st )𝑅) = ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩)
2		fo1st 8033	. . . . . . . 8 ⊢ 1st :V–onto→V
3		fofn 6823	. . . . . . . 8 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ 1st Fn V
5		opex 5475	. . . . . . 7 ⊢ ⟨𝑌, 𝑅⟩ ∈ V
6		fvco2 7006	. . . . . . 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 2763	. . . . 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 5780	. . . . . . 7 ⊢ (𝑌𝑊𝑅 → (𝑌 ∈ V ∧ 𝑅 ∈ V))
11		op1stg 8025	. . . . . . 7 ⊢ ((𝑌 ∈ V ∧ 𝑅 ∈ V) → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
12	10, 11	syl 17	. . . . . 6 ⊢ (𝑌𝑊𝑅 → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
13	12	fveq2d 6911	. . . . 5 ⊢ (𝑌𝑊𝑅 → (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)) = (𝐹‘𝑌))
14	8, 13	eqtrid 2787	. . . 4 ⊢ (𝑌𝑊𝑅 → (𝑌(𝐹 ∘ 1st )𝑅) = (𝐹‘𝑌))
15	14	eleq1d 2824	. . 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 3482	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
18	17	cnvex 7948	. . . . . . . . . 10 ⊢ ◡𝑟 ∈ V
19	18	imaex 7937	. . . . . . . . 9 ⊢ (◡𝑟 “ {𝑦}) ∈ V
20		vex 3482	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
21	17	inex1 5323	. . . . . . . . . . . 12 ⊢ (𝑟 ∩ (𝑢 × 𝑢)) ∈ V
22	20, 21	opco1i 8149	. . . . . . . . . . 11 ⊢ (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹‘𝑢)
23		fveq2 6907	. . . . . . . . . . 11 ⊢ (𝑢 = (◡𝑟 “ {𝑦}) → (𝐹‘𝑢) = (𝐹‘(◡𝑟 “ {𝑦})))
24	22, 23	eqtrid 2787	. . . . . . . . . 10 ⊢ (𝑢 = (◡𝑟 “ {𝑦}) → (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹‘(◡𝑟 “ {𝑦})))
25	24	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑢 = (◡𝑟 “ {𝑦}) → ((𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))
26	19, 25	sbcie 3835	. . . . . . . 8 ⊢ ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦)
27	26	ralbii 3091	. . . . . . 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 5215	. . . 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 2766	. . 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 3482	. . . . 5 ⊢ 𝑥 ∈ V
34	33, 17	opco1i 8149	. . . 4 ⊢ (𝑥(𝐹 ∘ 1st )𝑟) = (𝐹‘𝑥)
35		simp1 1135	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ⊆ 𝐴)
36		velpw 4610	. . . . . . 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 2179	. . . . . . . 8 ⊢ (𝑟 We 𝑥 → ∃𝑟 𝑟 We 𝑥)
39	38	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → ∃𝑟 𝑟 We 𝑥)
40		ween 10073	. . . . . . 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 4210	. . . . 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 2843	. . 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 10681	. 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 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  Vcvv 3478  [wsbc 3791   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605  {csn 4631  ⟨cop 4637  ∪ cuni 4912   class class class wbr 5148  {copab 5210   We wwe 5640   × cxp 5687  ^◡ccnv 5688  dom cdm 5689   “ cima 5692   ∘ ccom 5693   Fn wfn 6558  –onto→wfo 6561  ‘cfv 6563  (class class class)co 7431  1^st c1st 8011  cardccrd 9973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-en 8985  df-oi 9548  df-card 9977

This theorem is referenced by:  canth4  10685  canthnumlem  10686  canthp1lem2  10691