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Theorem elmapresaun 8895
Description: fresaun 6762 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Assertion
Ref Expression
elmapresaun ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) ∈ (𝐶m (𝐴𝐵)))

Proof of Theorem elmapresaun
StepHypRef Expression
1 elmapi 8864 . . 3 (𝐹 ∈ (𝐶m 𝐴) → 𝐹:𝐴𝐶)
2 elmapi 8864 . . 3 (𝐺 ∈ (𝐶m 𝐵) → 𝐺:𝐵𝐶)
3 id 22 . . 3 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
4 fresaun 6762 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
51, 2, 3, 4syl3an 1157 . 2 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
6 elmapex 8863 . . . . 5 (𝐹 ∈ (𝐶m 𝐴) → (𝐶 ∈ V ∧ 𝐴 ∈ V))
76simpld 493 . . . 4 (𝐹 ∈ (𝐶m 𝐴) → 𝐶 ∈ V)
873ad2ant1 1130 . . 3 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → 𝐶 ∈ V)
96simprd 494 . . . . 5 (𝐹 ∈ (𝐶m 𝐴) → 𝐴 ∈ V)
10 elmapex 8863 . . . . . 6 (𝐺 ∈ (𝐶m 𝐵) → (𝐶 ∈ V ∧ 𝐵 ∈ V))
1110simprd 494 . . . . 5 (𝐺 ∈ (𝐶m 𝐵) → 𝐵 ∈ V)
12 unexg 7748 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
139, 11, 12syl2an 594 . . . 4 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵)) → (𝐴𝐵) ∈ V)
14133adant3 1129 . . 3 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐴𝐵) ∈ V)
158, 14elmapd 8855 . 2 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ∈ (𝐶m (𝐴𝐵)) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶))
165, 15mpbird 256 1 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺) ∈ (𝐶m (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3463  cun 3938  cin 3939  cres 5674  wf 6538  (class class class)co 7415  m cmap 8841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-map 8843
This theorem is referenced by:  satfv1lem  35028  diophin  42256  eldioph4b  42295  diophren  42297
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