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Theorem elmapresaun 8919
Description: fresaun 6780 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 8888 . . 3 (𝐹 ∈ (𝐶m 𝐴) → 𝐹:𝐴𝐶)
2 elmapi 8888 . . 3 (𝐺 ∈ (𝐶m 𝐵) → 𝐺:𝐵𝐶)
3 id 22 . . 3 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
4 fresaun 6780 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
51, 2, 3, 4syl3an 1159 . 2 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
6 elmapex 8887 . . . . 5 (𝐹 ∈ (𝐶m 𝐴) → (𝐶 ∈ V ∧ 𝐴 ∈ V))
76simpld 494 . . . 4 (𝐹 ∈ (𝐶m 𝐴) → 𝐶 ∈ V)
873ad2ant1 1132 . . 3 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → 𝐶 ∈ V)
96simprd 495 . . . . 5 (𝐹 ∈ (𝐶m 𝐴) → 𝐴 ∈ V)
10 elmapex 8887 . . . . . 6 (𝐺 ∈ (𝐶m 𝐵) → (𝐶 ∈ V ∧ 𝐵 ∈ V))
1110simprd 495 . . . . 5 (𝐺 ∈ (𝐶m 𝐵) → 𝐵 ∈ V)
12 unexg 7762 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
139, 11, 12syl2an 596 . . . 4 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵)) → (𝐴𝐵) ∈ V)
14133adant3 1131 . . 3 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐴𝐵) ∈ V)
158, 14elmapd 8879 . 2 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ∈ (𝐶m (𝐴𝐵)) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶))
165, 15mpbird 257 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 1086   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  cin 3962  cres 5691  wf 6559  (class class class)co 7431  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by:  satfv1lem  35347  diophin  42760  eldioph4b  42799  diophren  42801
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