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Theorem ovmpt2s 6981
Description: Value of a function given by the maps-to notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)
Hypothesis
Ref Expression
ovmpt2s.3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ovmpt2s ((𝐴𝐶𝐵𝐷𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉) → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ovmpt2s
StepHypRef Expression
1 elex 3364 . . 3 (𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉𝐴 / 𝑥𝐵 / 𝑦𝑅 ∈ V)
2 nfcv 2906 . . . . 5 𝑥𝐴
3 nfcv 2906 . . . . 5 𝑦𝐴
4 nfcv 2906 . . . . 5 𝑦𝐵
5 nfcsb1v 3706 . . . . . . 7 𝑥𝐴 / 𝑥𝑅
65nfel1 2921 . . . . . 6 𝑥𝐴 / 𝑥𝑅 ∈ V
7 ovmpt2s.3 . . . . . . . . 9 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
8 nfmpt21 6919 . . . . . . . . 9 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
97, 8nfcxfr 2904 . . . . . . . 8 𝑥𝐹
10 nfcv 2906 . . . . . . . 8 𝑥𝑦
112, 9, 10nfov 6871 . . . . . . 7 𝑥(𝐴𝐹𝑦)
1211, 5nfeq 2918 . . . . . 6 𝑥(𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅
136, 12nfim 1995 . . . . 5 𝑥(𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅)
14 nfcsb1v 3706 . . . . . . 7 𝑦𝐵 / 𝑦𝐴 / 𝑥𝑅
1514nfel1 2921 . . . . . 6 𝑦𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V
16 nfmpt22 6920 . . . . . . . . 9 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
177, 16nfcxfr 2904 . . . . . . . 8 𝑦𝐹
183, 17, 4nfov 6871 . . . . . . 7 𝑦(𝐴𝐹𝐵)
1918, 14nfeq 2918 . . . . . 6 𝑦(𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅
2015, 19nfim 1995 . . . . 5 𝑦(𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅)
21 csbeq1a 3699 . . . . . . 7 (𝑥 = 𝐴𝑅 = 𝐴 / 𝑥𝑅)
2221eleq1d 2828 . . . . . 6 (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐴 / 𝑥𝑅 ∈ V))
23 oveq1 6848 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
2423, 21eqeq12d 2779 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅))
2522, 24imbi12d 335 . . . . 5 (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅)))
26 csbeq1a 3699 . . . . . . 7 (𝑦 = 𝐵𝐴 / 𝑥𝑅 = 𝐵 / 𝑦𝐴 / 𝑥𝑅)
2726eleq1d 2828 . . . . . 6 (𝑦 = 𝐵 → (𝐴 / 𝑥𝑅 ∈ V ↔ 𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V))
28 oveq2 6849 . . . . . . 7 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
2928, 26eqeq12d 2779 . . . . . 6 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅 ↔ (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅))
3027, 29imbi12d 335 . . . . 5 (𝑦 = 𝐵 → ((𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅) ↔ (𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅)))
317ovmpt4g 6980 . . . . . 6 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32313expia 1150 . . . . 5 ((𝑥𝐶𝑦𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 3424 . . . 4 ((𝐴𝐶𝐵𝐷) → (𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅))
34 csbcom 4154 . . . . 5 𝐴 / 𝑥𝐵 / 𝑦𝑅 = 𝐵 / 𝑦𝐴 / 𝑥𝑅
3534eleq1i 2834 . . . 4 (𝐴 / 𝑥𝐵 / 𝑦𝑅 ∈ V ↔ 𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V)
3634eqeq2i 2776 . . . 4 ((𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅 ↔ (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅)
3733, 35, 363imtr4g 287 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴 / 𝑥𝐵 / 𝑦𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅))
381, 37syl5 34 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉 → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅))
39383impia 1145 1 ((𝐴𝐶𝐵𝐷𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉) → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  Vcvv 3349  csb 3690  (class class class)co 6841  cmpt2 6843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-iota 6030  df-fun 6069  df-fv 6075  df-ov 6844  df-oprab 6845  df-mpt2 6846
This theorem is referenced by:  finxpreclem2  33592
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