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Theorem wdom2d 9032
Description: Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 5157). (Contributed by Stefan O'Rear, 13-Feb-2015.)
Hypotheses
Ref Expression
wdom2d.a (𝜑𝐴𝑉)
wdom2d.b (𝜑𝐵𝑊)
wdom2d.o ((𝜑𝑥𝐴) → ∃𝑦𝐵 𝑥 = 𝑋)
Assertion
Ref Expression
wdom2d (𝜑𝐴* 𝐵)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑋   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   𝑋(𝑦)

Proof of Theorem wdom2d
Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wdom2d.b . . . . . 6 (𝜑𝐵𝑊)
2 rabexg 5201 . . . . . 6 (𝐵𝑊 → {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ∈ V)
31, 2syl 17 . . . . 5 (𝜑 → {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ∈ V)
4 wdom2d.a . . . . 5 (𝜑𝐴𝑉)
53, 4xpexd 7458 . . . 4 (𝜑 → ({𝑧𝐵𝑧 / 𝑦𝑋𝐴} × 𝐴) ∈ V)
6 csbeq1 3834 . . . . . . . . . 10 (𝑧 = 𝑤𝑧 / 𝑦𝑋 = 𝑤 / 𝑦𝑋)
76eleq1d 2877 . . . . . . . . 9 (𝑧 = 𝑤 → (𝑧 / 𝑦𝑋𝐴𝑤 / 𝑦𝑋𝐴))
87elrab 3631 . . . . . . . 8 (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↔ (𝑤𝐵𝑤 / 𝑦𝑋𝐴))
98simprbi 500 . . . . . . 7 (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} → 𝑤 / 𝑦𝑋𝐴)
109adantl 485 . . . . . 6 ((𝜑𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}) → 𝑤 / 𝑦𝑋𝐴)
1110fmpttd 6860 . . . . 5 (𝜑 → (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋):{𝑧𝐵𝑧 / 𝑦𝑋𝐴}⟶𝐴)
12 fssxp 6512 . . . . 5 ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋):{𝑧𝐵𝑧 / 𝑦𝑋𝐴}⟶𝐴 → (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋) ⊆ ({𝑧𝐵𝑧 / 𝑦𝑋𝐴} × 𝐴))
1311, 12syl 17 . . . 4 (𝜑 → (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋) ⊆ ({𝑧𝐵𝑧 / 𝑦𝑋𝐴} × 𝐴))
145, 13ssexd 5195 . . 3 (𝜑 → (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋) ∈ V)
15 wdom2d.o . . . . . . . 8 ((𝜑𝑥𝐴) → ∃𝑦𝐵 𝑥 = 𝑋)
16 eleq1 2880 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
1716biimpcd 252 . . . . . . . . . . 11 (𝑥𝐴 → (𝑥 = 𝑋𝑋𝐴))
1817ancrd 555 . . . . . . . . . 10 (𝑥𝐴 → (𝑥 = 𝑋 → (𝑋𝐴𝑥 = 𝑋)))
1918adantl 485 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑥 = 𝑋 → (𝑋𝐴𝑥 = 𝑋)))
2019reximdv 3235 . . . . . . . 8 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝑥 = 𝑋 → ∃𝑦𝐵 (𝑋𝐴𝑥 = 𝑋)))
2115, 20mpd 15 . . . . . . 7 ((𝜑𝑥𝐴) → ∃𝑦𝐵 (𝑋𝐴𝑥 = 𝑋))
22 nfv 1915 . . . . . . . 8 𝑣(𝑋𝐴𝑥 = 𝑋)
23 nfcsb1v 3855 . . . . . . . . . 10 𝑦𝑣 / 𝑦𝑋
2423nfel1 2974 . . . . . . . . 9 𝑦𝑣 / 𝑦𝑋𝐴
2523nfeq2 2975 . . . . . . . . 9 𝑦 𝑥 = 𝑣 / 𝑦𝑋
2624, 25nfan 1900 . . . . . . . 8 𝑦(𝑣 / 𝑦𝑋𝐴𝑥 = 𝑣 / 𝑦𝑋)
27 csbeq1a 3845 . . . . . . . . . 10 (𝑦 = 𝑣𝑋 = 𝑣 / 𝑦𝑋)
2827eleq1d 2877 . . . . . . . . 9 (𝑦 = 𝑣 → (𝑋𝐴𝑣 / 𝑦𝑋𝐴))
2927eqeq2d 2812 . . . . . . . . 9 (𝑦 = 𝑣 → (𝑥 = 𝑋𝑥 = 𝑣 / 𝑦𝑋))
3028, 29anbi12d 633 . . . . . . . 8 (𝑦 = 𝑣 → ((𝑋𝐴𝑥 = 𝑋) ↔ (𝑣 / 𝑦𝑋𝐴𝑥 = 𝑣 / 𝑦𝑋)))
3122, 26, 30cbvrexw 3391 . . . . . . 7 (∃𝑦𝐵 (𝑋𝐴𝑥 = 𝑋) ↔ ∃𝑣𝐵 (𝑣 / 𝑦𝑋𝐴𝑥 = 𝑣 / 𝑦𝑋))
3221, 31sylib 221 . . . . . 6 ((𝜑𝑥𝐴) → ∃𝑣𝐵 (𝑣 / 𝑦𝑋𝐴𝑥 = 𝑣 / 𝑦𝑋))
33 csbeq1 3834 . . . . . . . . . . . . 13 (𝑧 = 𝑣𝑧 / 𝑦𝑋 = 𝑣 / 𝑦𝑋)
3433eleq1d 2877 . . . . . . . . . . . 12 (𝑧 = 𝑣 → (𝑧 / 𝑦𝑋𝐴𝑣 / 𝑦𝑋𝐴))
3534elrab 3631 . . . . . . . . . . 11 (𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↔ (𝑣𝐵𝑣 / 𝑦𝑋𝐴))
3635simprbi 500 . . . . . . . . . 10 (𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} → 𝑣 / 𝑦𝑋𝐴)
37 csbeq1 3834 . . . . . . . . . . 11 (𝑤 = 𝑣𝑤 / 𝑦𝑋 = 𝑣 / 𝑦𝑋)
38 eqid 2801 . . . . . . . . . . 11 (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋) = (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)
3937, 38fvmptg 6747 . . . . . . . . . 10 ((𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ∧ 𝑣 / 𝑦𝑋𝐴) → ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣) = 𝑣 / 𝑦𝑋)
4036, 39mpdan 686 . . . . . . . . 9 (𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} → ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣) = 𝑣 / 𝑦𝑋)
4140eqeq2d 2812 . . . . . . . 8 (𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} → (𝑥 = ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣) ↔ 𝑥 = 𝑣 / 𝑦𝑋))
4241rexbiia 3212 . . . . . . 7 (∃𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣) ↔ ∃𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = 𝑣 / 𝑦𝑋)
4334rexrab 3638 . . . . . . 7 (∃𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = 𝑣 / 𝑦𝑋 ↔ ∃𝑣𝐵 (𝑣 / 𝑦𝑋𝐴𝑥 = 𝑣 / 𝑦𝑋))
4442, 43bitri 278 . . . . . 6 (∃𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣) ↔ ∃𝑣𝐵 (𝑣 / 𝑦𝑋𝐴𝑥 = 𝑣 / 𝑦𝑋))
4532, 44sylibr 237 . . . . 5 ((𝜑𝑥𝐴) → ∃𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣))
4645ralrimiva 3152 . . . 4 (𝜑 → ∀𝑥𝐴𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣))
47 dffo3 6849 . . . 4 ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋):{𝑧𝐵𝑧 / 𝑦𝑋𝐴}–onto𝐴 ↔ ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋):{𝑧𝐵𝑧 / 𝑦𝑋𝐴}⟶𝐴 ∧ ∀𝑥𝐴𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣)))
4811, 46, 47sylanbrc 586 . . 3 (𝜑 → (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋):{𝑧𝐵𝑧 / 𝑦𝑋𝐴}–onto𝐴)
49 fowdom 9023 . . 3 (((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋) ∈ V ∧ (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋):{𝑧𝐵𝑧 / 𝑦𝑋𝐴}–onto𝐴) → 𝐴* {𝑧𝐵𝑧 / 𝑦𝑋𝐴})
5014, 48, 49syl2anc 587 . 2 (𝜑𝐴* {𝑧𝐵𝑧 / 𝑦𝑋𝐴})
51 ssrab2 4010 . . . 4 {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ⊆ 𝐵
52 ssdomg 8542 . . . 4 (𝐵𝑊 → ({𝑧𝐵𝑧 / 𝑦𝑋𝐴} ⊆ 𝐵 → {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ≼ 𝐵))
5351, 52mpi 20 . . 3 (𝐵𝑊 → {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ≼ 𝐵)
54 domwdom 9026 . . 3 ({𝑧𝐵𝑧 / 𝑦𝑋𝐴} ≼ 𝐵 → {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ≼* 𝐵)
551, 53, 543syl 18 . 2 (𝜑 → {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ≼* 𝐵)
56 wdomtr 9027 . 2 ((𝐴* {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ∧ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ≼* 𝐵) → 𝐴* 𝐵)
5750, 55, 56syl2anc 587 1 (𝜑𝐴* 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  csb 3831  wss 3884   class class class wbr 5033  cmpt 5113   × cxp 5521  wf 6324  ontowfo 6326  cfv 6328  cdom 8494  * cwdom 9016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-wdom 9017
This theorem is referenced by:  wdomd  9033  brwdom3  9034  unwdomg  9036  xpwdomg  9037  wdom2d2  39963
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