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Theorem ovmpos 5946
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
ovmpos.3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ovmpos ((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 ovmpos
StepHypRef Expression
1 elex 2723 . . 3 (𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉𝐴 / 𝑥𝐵 / 𝑦𝑅 ∈ V)
2 nfcv 2299 . . . . 5 𝑥𝐴
3 nfcv 2299 . . . . 5 𝑦𝐴
4 nfcv 2299 . . . . 5 𝑦𝐵
5 nfcsb1v 3064 . . . . . . 7 𝑥𝐴 / 𝑥𝑅
65nfel1 2310 . . . . . 6 𝑥𝐴 / 𝑥𝑅 ∈ V
7 ovmpos.3 . . . . . . . . 9 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
8 nfmpo1 5890 . . . . . . . . 9 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
97, 8nfcxfr 2296 . . . . . . . 8 𝑥𝐹
10 nfcv 2299 . . . . . . . 8 𝑥𝑦
112, 9, 10nfov 5853 . . . . . . 7 𝑥(𝐴𝐹𝑦)
1211, 5nfeq 2307 . . . . . 6 𝑥(𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅
136, 12nfim 1552 . . . . 5 𝑥(𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅)
14 nfcsb1v 3064 . . . . . . 7 𝑦𝐵 / 𝑦𝐴 / 𝑥𝑅
1514nfel1 2310 . . . . . 6 𝑦𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V
16 nfmpo2 5891 . . . . . . . . 9 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
177, 16nfcxfr 2296 . . . . . . . 8 𝑦𝐹
183, 17, 4nfov 5853 . . . . . . 7 𝑦(𝐴𝐹𝐵)
1918, 14nfeq 2307 . . . . . 6 𝑦(𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅
2015, 19nfim 1552 . . . . 5 𝑦(𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅)
21 csbeq1a 3040 . . . . . . 7 (𝑥 = 𝐴𝑅 = 𝐴 / 𝑥𝑅)
2221eleq1d 2226 . . . . . 6 (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐴 / 𝑥𝑅 ∈ V))
23 oveq1 5833 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
2423, 21eqeq12d 2172 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅))
2522, 24imbi12d 233 . . . . 5 (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅)))
26 csbeq1a 3040 . . . . . . 7 (𝑦 = 𝐵𝐴 / 𝑥𝑅 = 𝐵 / 𝑦𝐴 / 𝑥𝑅)
2726eleq1d 2226 . . . . . 6 (𝑦 = 𝐵 → (𝐴 / 𝑥𝑅 ∈ V ↔ 𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V))
28 oveq2 5834 . . . . . . 7 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
2928, 26eqeq12d 2172 . . . . . 6 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅 ↔ (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅))
3027, 29imbi12d 233 . . . . 5 (𝑦 = 𝐵 → ((𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅) ↔ (𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅)))
317ovmpt4g 5945 . . . . . 6 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32313expia 1187 . . . . 5 ((𝑥𝐶𝑦𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 2779 . . . 4 ((𝐴𝐶𝐵𝐷) → (𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅))
34 csbcomg 3054 . . . . 5 ((𝐴𝐶𝐵𝐷) → 𝐴 / 𝑥𝐵 / 𝑦𝑅 = 𝐵 / 𝑦𝐴 / 𝑥𝑅)
3534eleq1d 2226 . . . 4 ((𝐴𝐶𝐵𝐷) → (𝐴 / 𝑥𝐵 / 𝑦𝑅 ∈ V ↔ 𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V))
3634eqeq2d 2169 . . . 4 ((𝐴𝐶𝐵𝐷) → ((𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅 ↔ (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅))
3733, 35, 363imtr4d 202 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴 / 𝑥𝐵 / 𝑦𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅))
381, 37syl5 32 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉 → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅))
39383impia 1182 1 ((𝐴𝐶𝐵𝐷𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉) → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963   = wceq 1335  wcel 2128  Vcvv 2712  csb 3031  (class class class)co 5826  cmpo 5828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-setind 4498
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-iota 5137  df-fun 5174  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831
This theorem is referenced by: (None)
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