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Theorem ovmpt2s 5566
 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 2560 . . 3 (A / xB / y𝑅 𝑉A / xB / y𝑅 V)
2 nfcv 2175 . . . . 5 xA
3 nfcv 2175 . . . . 5 yA
4 nfcv 2175 . . . . 5 yB
5 nfcsb1v 2876 . . . . . . 7 xA / x𝑅
65nfel1 2185 . . . . . 6 xA / x𝑅 V
7 ovmpt2s.3 . . . . . . . . 9 𝐹 = (x 𝐶, y 𝐷𝑅)
8 nfmpt21 5513 . . . . . . . . 9 x(x 𝐶, y 𝐷𝑅)
97, 8nfcxfr 2172 . . . . . . . 8 x𝐹
10 nfcv 2175 . . . . . . . 8 xy
112, 9, 10nfov 5478 . . . . . . 7 x(A𝐹y)
1211, 5nfeq 2182 . . . . . 6 x(A𝐹y) = A / x𝑅
136, 12nfim 1461 . . . . 5 x(A / x𝑅 V → (A𝐹y) = A / x𝑅)
14 nfcsb1v 2876 . . . . . . 7 yB / yA / x𝑅
1514nfel1 2185 . . . . . 6 yB / yA / x𝑅 V
16 nfmpt22 5514 . . . . . . . . 9 y(x 𝐶, y 𝐷𝑅)
177, 16nfcxfr 2172 . . . . . . . 8 y𝐹
183, 17, 4nfov 5478 . . . . . . 7 y(A𝐹B)
1918, 14nfeq 2182 . . . . . 6 y(A𝐹B) = B / yA / x𝑅
2015, 19nfim 1461 . . . . 5 y(B / yA / x𝑅 V → (A𝐹B) = B / yA / x𝑅)
21 csbeq1a 2854 . . . . . . 7 (x = A𝑅 = A / x𝑅)
2221eleq1d 2103 . . . . . 6 (x = A → (𝑅 V ↔ A / x𝑅 V))
23 oveq1 5462 . . . . . . 7 (x = A → (x𝐹y) = (A𝐹y))
2423, 21eqeq12d 2051 . . . . . 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 2854 . . . . . . 7 (y = BA / x𝑅 = B / yA / x𝑅)
2726eleq1d 2103 . . . . . 6 (y = B → (A / x𝑅 V ↔ B / yA / x𝑅 V))
28 oveq2 5463 . . . . . . 7 (y = B → (A𝐹y) = (A𝐹B))
2928, 26eqeq12d 2051 . . . . . 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 5565 . . . . . 6 ((x 𝐶 y 𝐷 𝑅 V) → (x𝐹y) = 𝑅)
32313expia 1105 . . . . 5 ((x 𝐶 y 𝐷) → (𝑅 V → (x𝐹y) = 𝑅))
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 2614 . . . 4 ((A 𝐶 B 𝐷) → (B / yA / x𝑅 V → (A𝐹B) = B / yA / x𝑅))
34 csbcomg 2867 . . . . 5 ((A 𝐶 B 𝐷) → A / xB / y𝑅 = B / yA / x𝑅)
3534eleq1d 2103 . . . 4 ((A 𝐶 B 𝐷) → (A / xB / y𝑅 V ↔ B / yA / x𝑅 V))
3634eqeq2d 2048 . . . 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 1100 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 884   = wceq 1242   ∈ wcel 1390  Vcvv 2551  ⦋csb 2846  (class class class)co 5455   ↦ cmpt2 5457 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460 This theorem is referenced by: (None)
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