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Theorem wdom2d 9524
Description: Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 5246). (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 5292 . . . . . 6 (𝐵𝑊 → {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ∈ V)
31, 2syl 17 . . . . 5 (𝜑 → {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ∈ V)
4 wdom2d.a . . . . 5 (𝜑𝐴𝑉)
53, 4xpexd 7689 . . . 4 (𝜑 → ({𝑧𝐵𝑧 / 𝑦𝑋𝐴} × 𝐴) ∈ V)
6 csbeq1 3862 . . . . . . . . . 10 (𝑧 = 𝑤𝑧 / 𝑦𝑋 = 𝑤 / 𝑦𝑋)
76eleq1d 2819 . . . . . . . . 9 (𝑧 = 𝑤 → (𝑧 / 𝑦𝑋𝐴𝑤 / 𝑦𝑋𝐴))
87elrab 3649 . . . . . . . 8 (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↔ (𝑤𝐵𝑤 / 𝑦𝑋𝐴))
98simprbi 498 . . . . . . 7 (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} → 𝑤 / 𝑦𝑋𝐴)
109adantl 483 . . . . . 6 ((𝜑𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}) → 𝑤 / 𝑦𝑋𝐴)
1110fmpttd 7067 . . . . 5 (𝜑 → (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋):{𝑧𝐵𝑧 / 𝑦𝑋𝐴}⟶𝐴)
12 fssxp 6700 . . . . 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 5285 . . 3 (𝜑 → (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋) ∈ V)
15 wdom2d.o . . . . . . . 8 ((𝜑𝑥𝐴) → ∃𝑦𝐵 𝑥 = 𝑋)
16 eleq1 2822 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
1716biimpcd 249 . . . . . . . . . . 11 (𝑥𝐴 → (𝑥 = 𝑋𝑋𝐴))
1817ancrd 553 . . . . . . . . . 10 (𝑥𝐴 → (𝑥 = 𝑋 → (𝑋𝐴𝑥 = 𝑋)))
1918adantl 483 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑥 = 𝑋 → (𝑋𝐴𝑥 = 𝑋)))
2019reximdv 3164 . . . . . . . 8 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝑥 = 𝑋 → ∃𝑦𝐵 (𝑋𝐴𝑥 = 𝑋)))
2115, 20mpd 15 . . . . . . 7 ((𝜑𝑥𝐴) → ∃𝑦𝐵 (𝑋𝐴𝑥 = 𝑋))
22 nfv 1918 . . . . . . . 8 𝑣(𝑋𝐴𝑥 = 𝑋)
23 nfcsb1v 3884 . . . . . . . . . 10 𝑦𝑣 / 𝑦𝑋
2423nfel1 2920 . . . . . . . . 9 𝑦𝑣 / 𝑦𝑋𝐴
2523nfeq2 2921 . . . . . . . . 9 𝑦 𝑥 = 𝑣 / 𝑦𝑋
2624, 25nfan 1903 . . . . . . . 8 𝑦(𝑣 / 𝑦𝑋𝐴𝑥 = 𝑣 / 𝑦𝑋)
27 csbeq1a 3873 . . . . . . . . . 10 (𝑦 = 𝑣𝑋 = 𝑣 / 𝑦𝑋)
2827eleq1d 2819 . . . . . . . . 9 (𝑦 = 𝑣 → (𝑋𝐴𝑣 / 𝑦𝑋𝐴))
2927eqeq2d 2744 . . . . . . . . 9 (𝑦 = 𝑣 → (𝑥 = 𝑋𝑥 = 𝑣 / 𝑦𝑋))
3028, 29anbi12d 632 . . . . . . . 8 (𝑦 = 𝑣 → ((𝑋𝐴𝑥 = 𝑋) ↔ (𝑣 / 𝑦𝑋𝐴𝑥 = 𝑣 / 𝑦𝑋)))
3122, 26, 30cbvrexw 3289 . . . . . . 7 (∃𝑦𝐵 (𝑋𝐴𝑥 = 𝑋) ↔ ∃𝑣𝐵 (𝑣 / 𝑦𝑋𝐴𝑥 = 𝑣 / 𝑦𝑋))
3221, 31sylib 217 . . . . . 6 ((𝜑𝑥𝐴) → ∃𝑣𝐵 (𝑣 / 𝑦𝑋𝐴𝑥 = 𝑣 / 𝑦𝑋))
33 csbeq1 3862 . . . . . . . . . . . . 13 (𝑧 = 𝑣𝑧 / 𝑦𝑋 = 𝑣 / 𝑦𝑋)
3433eleq1d 2819 . . . . . . . . . . . 12 (𝑧 = 𝑣 → (𝑧 / 𝑦𝑋𝐴𝑣 / 𝑦𝑋𝐴))
3534elrab 3649 . . . . . . . . . . 11 (𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↔ (𝑣𝐵𝑣 / 𝑦𝑋𝐴))
3635simprbi 498 . . . . . . . . . 10 (𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} → 𝑣 / 𝑦𝑋𝐴)
37 csbeq1 3862 . . . . . . . . . . 11 (𝑤 = 𝑣𝑤 / 𝑦𝑋 = 𝑣 / 𝑦𝑋)
38 eqid 2733 . . . . . . . . . . 11 (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋) = (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)
3937, 38fvmptg 6950 . . . . . . . . . 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 2744 . . . . . . . 8 (𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} → (𝑥 = ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣) ↔ 𝑥 = 𝑣 / 𝑦𝑋))
4241rexbiia 3092 . . . . . . 7 (∃𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣) ↔ ∃𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = 𝑣 / 𝑦𝑋)
4334rexrab 3658 . . . . . . 7 (∃𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = 𝑣 / 𝑦𝑋 ↔ ∃𝑣𝐵 (𝑣 / 𝑦𝑋𝐴𝑥 = 𝑣 / 𝑦𝑋))
4442, 43bitri 275 . . . . . 6 (∃𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣) ↔ ∃𝑣𝐵 (𝑣 / 𝑦𝑋𝐴𝑥 = 𝑣 / 𝑦𝑋))
4532, 44sylibr 233 . . . . 5 ((𝜑𝑥𝐴) → ∃𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣))
4645ralrimiva 3140 . . . 4 (𝜑 → ∀𝑥𝐴𝑣 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴}𝑥 = ((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋)‘𝑣))
47 dffo3 7056 . . . 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 584 . . 3 (𝜑 → (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋):{𝑧𝐵𝑧 / 𝑦𝑋𝐴}–onto𝐴)
49 fowdom 9515 . . 3 (((𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋) ∈ V ∧ (𝑤 ∈ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ↦ 𝑤 / 𝑦𝑋):{𝑧𝐵𝑧 / 𝑦𝑋𝐴}–onto𝐴) → 𝐴* {𝑧𝐵𝑧 / 𝑦𝑋𝐴})
5014, 48, 49syl2anc 585 . 2 (𝜑𝐴* {𝑧𝐵𝑧 / 𝑦𝑋𝐴})
51 ssrab2 4041 . . . 4 {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ⊆ 𝐵
52 ssdomg 8946 . . . 4 (𝐵𝑊 → ({𝑧𝐵𝑧 / 𝑦𝑋𝐴} ⊆ 𝐵 → {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ≼ 𝐵))
5351, 52mpi 20 . . 3 (𝐵𝑊 → {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ≼ 𝐵)
54 domwdom 9518 . . 3 ({𝑧𝐵𝑧 / 𝑦𝑋𝐴} ≼ 𝐵 → {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ≼* 𝐵)
551, 53, 543syl 18 . 2 (𝜑 → {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ≼* 𝐵)
56 wdomtr 9519 . 2 ((𝐴* {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ∧ {𝑧𝐵𝑧 / 𝑦𝑋𝐴} ≼* 𝐵) → 𝐴* 𝐵)
5750, 55, 56syl2anc 585 1 (𝜑𝐴* 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070  {crab 3406  Vcvv 3447  csb 3859  wss 3914   class class class wbr 5109  cmpt 5192   × cxp 5635  wf 6496  ontowfo 6498  cfv 6500  cdom 8887  * cwdom 9508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-en 8890  df-dom 8891  df-sdom 8892  df-wdom 9509
This theorem is referenced by:  wdomd  9525  brwdom3  9526  unwdomg  9528  xpwdomg  9529  wdom2d2  41406
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