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Theorem ovmpt2s 5624
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 2566 . . 3 (𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉𝐴 / 𝑥𝐵 / 𝑦𝑅 ∈ V)
2 nfcv 2178 . . . . 5 𝑥𝐴
3 nfcv 2178 . . . . 5 𝑦𝐴
4 nfcv 2178 . . . . 5 𝑦𝐵
5 nfcsb1v 2882 . . . . . . 7 𝑥𝐴 / 𝑥𝑅
65nfel1 2188 . . . . . 6 𝑥𝐴 / 𝑥𝑅 ∈ V
7 ovmpt2s.3 . . . . . . . . 9 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
8 nfmpt21 5571 . . . . . . . . 9 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
97, 8nfcxfr 2175 . . . . . . . 8 𝑥𝐹
10 nfcv 2178 . . . . . . . 8 𝑥𝑦
112, 9, 10nfov 5535 . . . . . . 7 𝑥(𝐴𝐹𝑦)
1211, 5nfeq 2185 . . . . . 6 𝑥(𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅
136, 12nfim 1464 . . . . 5 𝑥(𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅)
14 nfcsb1v 2882 . . . . . . 7 𝑦𝐵 / 𝑦𝐴 / 𝑥𝑅
1514nfel1 2188 . . . . . 6 𝑦𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V
16 nfmpt22 5572 . . . . . . . . 9 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
177, 16nfcxfr 2175 . . . . . . . 8 𝑦𝐹
183, 17, 4nfov 5535 . . . . . . 7 𝑦(𝐴𝐹𝐵)
1918, 14nfeq 2185 . . . . . 6 𝑦(𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅
2015, 19nfim 1464 . . . . 5 𝑦(𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅)
21 csbeq1a 2860 . . . . . . 7 (𝑥 = 𝐴𝑅 = 𝐴 / 𝑥𝑅)
2221eleq1d 2106 . . . . . 6 (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐴 / 𝑥𝑅 ∈ V))
23 oveq1 5519 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
2423, 21eqeq12d 2054 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅))
2522, 24imbi12d 223 . . . . 5 (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅)))
26 csbeq1a 2860 . . . . . . 7 (𝑦 = 𝐵𝐴 / 𝑥𝑅 = 𝐵 / 𝑦𝐴 / 𝑥𝑅)
2726eleq1d 2106 . . . . . 6 (𝑦 = 𝐵 → (𝐴 / 𝑥𝑅 ∈ V ↔ 𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V))
28 oveq2 5520 . . . . . . 7 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
2928, 26eqeq12d 2054 . . . . . 6 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅 ↔ (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅))
3027, 29imbi12d 223 . . . . 5 (𝑦 = 𝐵 → ((𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅) ↔ (𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅)))
317ovmpt4g 5623 . . . . . 6 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32313expia 1106 . . . . 5 ((𝑥𝐶𝑦𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 2620 . . . 4 ((𝐴𝐶𝐵𝐷) → (𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅))
34 csbcomg 2873 . . . . 5 ((𝐴𝐶𝐵𝐷) → 𝐴 / 𝑥𝐵 / 𝑦𝑅 = 𝐵 / 𝑦𝐴 / 𝑥𝑅)
3534eleq1d 2106 . . . 4 ((𝐴𝐶𝐵𝐷) → (𝐴 / 𝑥𝐵 / 𝑦𝑅 ∈ V ↔ 𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V))
3634eqeq2d 2051 . . . 4 ((𝐴𝐶𝐵𝐷) → ((𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅 ↔ (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅))
3733, 35, 363imtr4d 192 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴 / 𝑥𝐵 / 𝑦𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅))
381, 37syl5 28 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉 → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅))
39383impia 1101 1 ((𝐴𝐶𝐵𝐷𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉) → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  w3a 885   = wceq 1243  wcel 1393  Vcvv 2557  csb 2852  (class class class)co 5512  cmpt2 5514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517
This theorem is referenced by: (None)
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