Theorem fpwwe 10068

Description: Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 8670 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 9456. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)

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 hint:   𝐴(𝑦)

Proof of Theorem fpwwe

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7159	. . . . . 6 ⊢ (𝑌(𝐹 ∘ 1st )𝑅) = ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩)
2		fo1st 7709	. . . . . . . 8 ⊢ 1st :V–onto→V
3		fofn 6592	. . . . . . . 8 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ 1st Fn V
5		opex 5356	. . . . . . 7 ⊢ ⟨𝑌, 𝑅⟩ ∈ V
6		fvco2 6758	. . . . . . 7 ⊢ ((1st Fn V ∧ ⟨𝑌, 𝑅⟩ ∈ V) → ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)))
7	4, 5, 6	mp2an 690	. . . . . 6 ⊢ ((𝐹 ∘ 1st )‘⟨𝑌, 𝑅⟩) = (𝐹‘(1st ‘⟨𝑌, 𝑅⟩))
8	1, 7	eqtri 2844	. . . . 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 5642	. . . . . . 7 ⊢ (𝑌𝑊𝑅 → (𝑌 ∈ V ∧ 𝑅 ∈ V))
11		op1stg 7701	. . . . . . 7 ⊢ ((𝑌 ∈ V ∧ 𝑅 ∈ V) → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
12	10, 11	syl 17	. . . . . 6 ⊢ (𝑌𝑊𝑅 → (1st ‘⟨𝑌, 𝑅⟩) = 𝑌)
13	12	fveq2d 6674	. . . . 5 ⊢ (𝑌𝑊𝑅 → (𝐹‘(1st ‘⟨𝑌, 𝑅⟩)) = (𝐹‘𝑌))
14	8, 13	syl5eq 2868	. . . 4 ⊢ (𝑌𝑊𝑅 → (𝑌(𝐹 ∘ 1st )𝑅) = (𝐹‘𝑌))
15	14	eleq1d 2897	. . 3 ⊢ (𝑌𝑊𝑅 → ((𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌 ↔ (𝐹‘𝑌) ∈ 𝑌))
16	15	pm5.32i 577	. 2 ⊢ ((𝑌𝑊𝑅 ∧ (𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌) ↔ (𝑌𝑊𝑅 ∧ (𝐹‘𝑌) ∈ 𝑌))
17		vex 3497	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
18	17	cnvex 7630	. . . . . . . . . 10 ⊢ ◡𝑟 ∈ V
19	18	imaex 7621	. . . . . . . . 9 ⊢ (◡𝑟 “ {𝑦}) ∈ V
20		vex 3497	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
21	17	inex1 5221	. . . . . . . . . . . 12 ⊢ (𝑟 ∩ (𝑢 × 𝑢)) ∈ V
22	20, 21	algrflem 7819	. . . . . . . . . . 11 ⊢ (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹‘𝑢)
23		fveq2 6670	. . . . . . . . . . 11 ⊢ (𝑢 = (◡𝑟 “ {𝑦}) → (𝐹‘𝑢) = (𝐹‘(◡𝑟 “ {𝑦})))
24	22, 23	syl5eq 2868	. . . . . . . . . 10 ⊢ (𝑢 = (◡𝑟 “ {𝑦}) → (𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = (𝐹‘(◡𝑟 “ {𝑦})))
25	24	eqeq1d 2823	. . . . . . . . 9 ⊢ (𝑢 = (◡𝑟 “ {𝑦}) → ((𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))
26	19, 25	sbcie 3812	. . . . . . . 8 ⊢ ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦)
27	26	ralbii 3165	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢(𝐹 ∘ 1st )(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦)
28	27	anbi2i 624	. . . . . 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 624	. . . . 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 5133	. . . 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 2847	. . 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 3497	. . . . 5 ⊢ 𝑥 ∈ V
34	33, 17	algrflem 7819	. . . 4 ⊢ (𝑥(𝐹 ∘ 1st )𝑟) = (𝐹‘𝑥)
35		simp1 1132	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ⊆ 𝐴)
36		velpw 4544	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
37	35, 36	sylibr 236	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
38		19.8a 2180	. . . . . . . 8 ⊢ (𝑟 We 𝑥 → ∃𝑟 𝑟 We 𝑥)
39	38	3ad2ant3 1131	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → ∃𝑟 𝑟 We 𝑥)
40		ween 9461	. . . . . . 7 ⊢ (𝑥 ∈ dom card ↔ ∃𝑟 𝑟 We 𝑥)
41	39, 40	sylibr 236	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ dom card)
42	37, 41	elind 4171	. . . . 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 594	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹‘𝑥) ∈ 𝐴)
45	34, 44	eqeltrid 2917	. . 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 10065	. 2 ⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌(𝐹 ∘ 1st )𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))
48	16, 47	syl5bbr 287	1 ⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹‘𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3138  Vcvv 3494  [wsbc 3772   ∩ cin 3935   ⊆ wss 3936  𝒫 cpw 4539  {csn 4567  ⟨cop 4573  ∪ cuni 4838   class class class wbr 5066  {copab 5128   We wwe 5513   × cxp 5553  ^◡ccnv 5554  dom cdm 5555   “ cima 5558   ∘ ccom 5559   Fn wfn 6350  –onto→wfo 6353  ‘cfv 6355  (class class class)co 7156  1^st c1st 7687  cardccrd 9364

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-1st 7689  df-wrecs 7947  df-recs 8008  df-en 8510  df-oi 8974  df-card 9368

This theorem is referenced by:  canth4  10069  canthnumlem  10070  canthp1lem2  10075