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Theorem ovmpt2s 5547
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 2543 . . 3 (A / xB / y𝑅 𝑉A / xB / y𝑅 V)
2 nfcv 2160 . . . . 5 xA
3 nfcv 2160 . . . . 5 yA
4 nfcv 2160 . . . . 5 yB
5 nfcsb1v 2859 . . . . . . 7 xA / x𝑅
65nfel1 2170 . . . . . 6 xA / x𝑅 V
7 ovmpt2s.3 . . . . . . . . 9 𝐹 = (x 𝐶, y 𝐷𝑅)
8 nfmpt21 5494 . . . . . . . . 9 x(x 𝐶, y 𝐷𝑅)
97, 8nfcxfr 2157 . . . . . . . 8 x𝐹
10 nfcv 2160 . . . . . . . 8 xy
112, 9, 10nfov 5459 . . . . . . 7 x(A𝐹y)
1211, 5nfeq 2167 . . . . . 6 x(A𝐹y) = A / x𝑅
136, 12nfim 1446 . . . . 5 x(A / x𝑅 V → (A𝐹y) = A / x𝑅)
14 nfcsb1v 2859 . . . . . . 7 yB / yA / x𝑅
1514nfel1 2170 . . . . . 6 yB / yA / x𝑅 V
16 nfmpt22 5495 . . . . . . . . 9 y(x 𝐶, y 𝐷𝑅)
177, 16nfcxfr 2157 . . . . . . . 8 y𝐹
183, 17, 4nfov 5459 . . . . . . 7 y(A𝐹B)
1918, 14nfeq 2167 . . . . . 6 y(A𝐹B) = B / yA / x𝑅
2015, 19nfim 1446 . . . . 5 y(B / yA / x𝑅 V → (A𝐹B) = B / yA / x𝑅)
21 csbeq1a 2837 . . . . . . 7 (x = A𝑅 = A / x𝑅)
2221eleq1d 2088 . . . . . 6 (x = A → (𝑅 V ↔ A / x𝑅 V))
23 oveq1 5443 . . . . . . 7 (x = A → (x𝐹y) = (A𝐹y))
2423, 21eqeq12d 2036 . . . . . 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 2837 . . . . . . 7 (y = BA / x𝑅 = B / yA / x𝑅)
2726eleq1d 2088 . . . . . 6 (y = B → (A / x𝑅 V ↔ B / yA / x𝑅 V))
28 oveq2 5444 . . . . . . 7 (y = B → (A𝐹y) = (A𝐹B))
2928, 26eqeq12d 2036 . . . . . 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 5546 . . . . . 6 ((x 𝐶 y 𝐷 𝑅 V) → (x𝐹y) = 𝑅)
32313expia 1092 . . . . 5 ((x 𝐶 y 𝐷) → (𝑅 V → (x𝐹y) = 𝑅))
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 2597 . . . 4 ((A 𝐶 B 𝐷) → (B / yA / x𝑅 V → (A𝐹B) = B / yA / x𝑅))
34 csbcomg 2850 . . . . 5 ((A 𝐶 B 𝐷) → A / xB / y𝑅 = B / yA / x𝑅)
3534eleq1d 2088 . . . 4 ((A 𝐶 B 𝐷) → (A / xB / y𝑅 V ↔ B / yA / x𝑅 V))
3634eqeq2d 2033 . . . 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 1087 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 873   = wceq 1228   wcel 1374  Vcvv 2535  csb 2829  (class class class)co 5436  cmpt2 5438
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441
This theorem is referenced by: (None)
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