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Theorem ovmpt2s 5536
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 = (x 𝐶, y 𝐷𝑅)
Assertion
Ref Expression
ovmpt2s ((A 𝐶 B 𝐷 A / xB / y𝑅 𝑉) → (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 2535 . . 3 (A / xB / y𝑅 𝑉A / xB / y𝑅 V)
2 nfcv 2152 . . . . 5 xA
3 nfcv 2152 . . . . 5 yA
4 nfcv 2152 . . . . 5 yB
5 nfcsb1v 2851 . . . . . . 7 xA / x𝑅
65nfel1 2162 . . . . . 6 xA / x𝑅 V
7 ovmpt2s.3 . . . . . . . . 9 𝐹 = (x 𝐶, y 𝐷𝑅)
8 nfmpt21 5483 . . . . . . . . 9 x(x 𝐶, y 𝐷𝑅)
97, 8nfcxfr 2149 . . . . . . . 8 x𝐹
10 nfcv 2152 . . . . . . . 8 xy
112, 9, 10nfov 5448 . . . . . . 7 x(A𝐹y)
1211, 5nfeq 2159 . . . . . 6 x(A𝐹y) = A / x𝑅
136, 12nfim 1438 . . . . 5 x(A / x𝑅 V → (A𝐹y) = A / x𝑅)
14 nfcsb1v 2851 . . . . . . 7 yB / yA / x𝑅
1514nfel1 2162 . . . . . 6 yB / yA / x𝑅 V
16 nfmpt22 5484 . . . . . . . . 9 y(x 𝐶, y 𝐷𝑅)
177, 16nfcxfr 2149 . . . . . . . 8 y𝐹
183, 17, 4nfov 5448 . . . . . . 7 y(A𝐹B)
1918, 14nfeq 2159 . . . . . 6 y(A𝐹B) = B / yA / x𝑅
2015, 19nfim 1438 . . . . 5 y(B / yA / x𝑅 V → (A𝐹B) = B / yA / x𝑅)
21 csbeq1a 2829 . . . . . . 7 (x = A𝑅 = A / x𝑅)
2221eleq1d 2080 . . . . . 6 (x = A → (𝑅 V ↔ A / x𝑅 V))
23 oveq1 5432 . . . . . . 7 (x = A → (x𝐹y) = (A𝐹y))
2423, 21eqeq12d 2028 . . . . . 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 2829 . . . . . . 7 (y = BA / x𝑅 = B / yA / x𝑅)
2726eleq1d 2080 . . . . . 6 (y = B → (A / x𝑅 V ↔ B / yA / x𝑅 V))
28 oveq2 5433 . . . . . . 7 (y = B → (A𝐹y) = (A𝐹B))
2928, 26eqeq12d 2028 . . . . . 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 5535 . . . . . 6 ((x 𝐶 y 𝐷 𝑅 V) → (x𝐹y) = 𝑅)
32313expia 1087 . . . . 5 ((x 𝐶 y 𝐷) → (𝑅 V → (x𝐹y) = 𝑅))
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 2589 . . . 4 ((A 𝐶 B 𝐷) → (B / yA / x𝑅 V → (A𝐹B) = B / yA / x𝑅))
34 csbcomg 2842 . . . . 5 ((A 𝐶 B 𝐷) → A / xB / y𝑅 = B / yA / x𝑅)
3534eleq1d 2080 . . . 4 ((A 𝐶 B 𝐷) → (A / xB / y𝑅 V ↔ B / yA / x𝑅 V))
3634eqeq2d 2025 . . . 4 ((A 𝐶 B 𝐷) → ((A𝐹B) = A / xB / y𝑅 ↔ (A𝐹B) = B / yA / x𝑅))
3733, 35, 363imtr4d 192 . . 3 ((A 𝐶 B 𝐷) → (A / xB / y𝑅 V → (A𝐹B) = A / xB / y𝑅))
381, 37syl5 28 . 2 ((A 𝐶 B 𝐷) → (A / xB / y𝑅 𝑉 → (A𝐹B) = A / xB / y𝑅))
39383impia 1082 1 ((A 𝐶 B 𝐷 A / xB / y𝑅 𝑉) → (A𝐹B) = A / xB / y𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 867   = wceq 1224   wcel 1367  Vcvv 2527  csb 2821  (class class class)co 5425  cmpt2 5427
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 529  ax-in2 530  ax-io 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908  ax-setind 4193
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-fal 1230  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ne 2180  df-ral 2281  df-rex 2282  df-v 2529  df-sbc 2734  df-csb 2822  df-dif 2889  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-uni 3545  df-br 3729  df-opab 3783  df-id 3994  df-xp 4267  df-rel 4268  df-cnv 4269  df-co 4270  df-dm 4271  df-iota 4783  df-fun 4820  df-fv 4826  df-ov 5428  df-oprab 5429  df-mpt2 5430
This theorem is referenced by: (None)
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