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Theorem ovmpos 7421
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 3450 . . 3 (𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉𝐴 / 𝑥𝐵 / 𝑦𝑅 ∈ V)
2 nfcv 2907 . . . . 5 𝑥𝐴
3 nfcv 2907 . . . . 5 𝑦𝐴
4 nfcv 2907 . . . . 5 𝑦𝐵
5 nfcsb1v 3857 . . . . . . 7 𝑥𝐴 / 𝑥𝑅
65nfel1 2923 . . . . . 6 𝑥𝐴 / 𝑥𝑅 ∈ V
7 ovmpos.3 . . . . . . . . 9 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
8 nfmpo1 7355 . . . . . . . . 9 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
97, 8nfcxfr 2905 . . . . . . . 8 𝑥𝐹
10 nfcv 2907 . . . . . . . 8 𝑥𝑦
112, 9, 10nfov 7305 . . . . . . 7 𝑥(𝐴𝐹𝑦)
1211, 5nfeq 2920 . . . . . 6 𝑥(𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅
136, 12nfim 1899 . . . . 5 𝑥(𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅)
14 nfcsb1v 3857 . . . . . . 7 𝑦𝐵 / 𝑦𝐴 / 𝑥𝑅
1514nfel1 2923 . . . . . 6 𝑦𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V
16 nfmpo2 7356 . . . . . . . . 9 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
177, 16nfcxfr 2905 . . . . . . . 8 𝑦𝐹
183, 17, 4nfov 7305 . . . . . . 7 𝑦(𝐴𝐹𝐵)
1918, 14nfeq 2920 . . . . . 6 𝑦(𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅
2015, 19nfim 1899 . . . . 5 𝑦(𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅)
21 csbeq1a 3846 . . . . . . 7 (𝑥 = 𝐴𝑅 = 𝐴 / 𝑥𝑅)
2221eleq1d 2823 . . . . . 6 (𝑥 = 𝐴 → (𝑅 ∈ V ↔ 𝐴 / 𝑥𝑅 ∈ V))
23 oveq1 7282 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
2423, 21eqeq12d 2754 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅))
2522, 24imbi12d 345 . . . . 5 (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅)))
26 csbeq1a 3846 . . . . . . 7 (𝑦 = 𝐵𝐴 / 𝑥𝑅 = 𝐵 / 𝑦𝐴 / 𝑥𝑅)
2726eleq1d 2823 . . . . . 6 (𝑦 = 𝐵 → (𝐴 / 𝑥𝑅 ∈ V ↔ 𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V))
28 oveq2 7283 . . . . . . 7 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
2928, 26eqeq12d 2754 . . . . . 6 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅 ↔ (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅))
3027, 29imbi12d 345 . . . . 5 (𝑦 = 𝐵 → ((𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝑦) = 𝐴 / 𝑥𝑅) ↔ (𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅)))
317ovmpt4g 7420 . . . . . 6 ((𝑥𝐶𝑦𝐷𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32313expia 1120 . . . . 5 ((𝑥𝐶𝑦𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 3515 . . . 4 ((𝐴𝐶𝐵𝐷) → (𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅))
34 csbcom 4351 . . . . 5 𝐴 / 𝑥𝐵 / 𝑦𝑅 = 𝐵 / 𝑦𝐴 / 𝑥𝑅
3534eleq1i 2829 . . . 4 (𝐴 / 𝑥𝐵 / 𝑦𝑅 ∈ V ↔ 𝐵 / 𝑦𝐴 / 𝑥𝑅 ∈ V)
3634eqeq2i 2751 . . . 4 ((𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅 ↔ (𝐴𝐹𝐵) = 𝐵 / 𝑦𝐴 / 𝑥𝑅)
3733, 35, 363imtr4g 296 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴 / 𝑥𝐵 / 𝑦𝑅 ∈ V → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅))
381, 37syl5 34 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉 → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅))
39383impia 1116 1 ((𝐴𝐶𝐵𝐷𝐴 / 𝑥𝐵 / 𝑦𝑅𝑉) → (𝐴𝐹𝐵) = 𝐴 / 𝑥𝐵 / 𝑦𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  Vcvv 3432  csb 3832  (class class class)co 7275  cmpo 7277
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280
This theorem is referenced by:  finxpreclem2  35561
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