Theorem ovmpt2g 5716

Description: Value of an operation given by a maps-to rule. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by set.mm contributors, 2-Oct-2007.) (Revised by set.mm contributors, 24-Jul-2012.)

Hypotheses

Ref	Expression
ovmpt2g.1	⊢ (x = A → R = G)
ovmpt2g.2	⊢ (y = B → G = S)
ovmpt2g.3	⊢ F = (x ∈ C, y ∈ D ↦ R)

Assertion

Ref	Expression
ovmpt2g	⊢ ((A ∈ C ∧ B ∈ D ∧ S ∈ H) → (AFB) = S)

Distinct variable groups:   x,y,A   y,B   x,C,y   x,D,y   x,G   y,S,x   x,B

Allowed substitution hints:   R(x,y)   F(x,y)   G(y)   H(x,y)

Proof of Theorem ovmpt2g

Step	Hyp	Ref	Expression
1		ovmpt2g.1	. . 3 ⊢ (x = A → R = G)
2		ovmpt2g.2	. . 3 ⊢ (y = B → G = S)
3	1, 2	sylan9eq 2405	. 2 ⊢ ((x = A ∧ y = B) → R = S)
4		ovmpt2g.3	. 2 ⊢ F = (x ∈ C, y ∈ D ↦ R)
5	3, 4	ovmpt2ga 5714	1 ⊢ ((A ∈ C ∧ B ∈ D ∧ S ∈ H) → (AFB) = S)

Syntax hints:   → wi 4   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  (class class class)co 5526   ↦ cmpt2 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fv 4796  df-ov 5527  df-oprab 5529  df-mpt2 5655

This theorem is referenced by:  ovmpt2  5717  cupvalg  5813  composevalg  5818  ovcross  5846  mapvalg  6010  pmvalg  6011  ovmuc  6131  ovce  6173