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Theorem elmapresaun 8433
Description: fresaun 6542 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 8417 . . 3 (𝐹 ∈ (𝐶m 𝐴) → 𝐹:𝐴𝐶)
2 elmapi 8417 . . 3 (𝐺 ∈ (𝐶m 𝐵) → 𝐺:𝐵𝐶)
3 id 22 . . 3 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
4 fresaun 6542 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
51, 2, 3, 4syl3an 1152 . 2 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
6 elmapex 8416 . . . . 5 (𝐹 ∈ (𝐶m 𝐴) → (𝐶 ∈ V ∧ 𝐴 ∈ V))
76simpld 495 . . . 4 (𝐹 ∈ (𝐶m 𝐴) → 𝐶 ∈ V)
873ad2ant1 1125 . . 3 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → 𝐶 ∈ V)
96simprd 496 . . . . 5 (𝐹 ∈ (𝐶m 𝐴) → 𝐴 ∈ V)
10 elmapex 8416 . . . . . 6 (𝐺 ∈ (𝐶m 𝐵) → (𝐶 ∈ V ∧ 𝐵 ∈ V))
1110simprd 496 . . . . 5 (𝐺 ∈ (𝐶m 𝐵) → 𝐵 ∈ V)
12 unexg 7461 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
139, 11, 12syl2an 595 . . . 4 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵)) → (𝐴𝐵) ∈ V)
14133adant3 1124 . . 3 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐴𝐵) ∈ V)
158, 14elmapd 8409 . 2 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ∈ (𝐶m (𝐴𝐵)) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶))
165, 15mpbird 258 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 1079   = wceq 1528  wcel 2105  Vcvv 3492  cun 3931  cin 3932  cres 5550  wf 6344  (class class class)co 7145  m cmap 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397
This theorem is referenced by:  satfv1lem  32506  diophin  39247  eldioph4b  39286  diophren  39288
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