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Theorem elmapresaun 36352
 Description: fresaun 5988 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 7765 . . 3 (𝐹 ∈ (𝐶𝑚 𝐴) → 𝐹:𝐴𝐶)
2 elmapi 7765 . . 3 (𝐺 ∈ (𝐶𝑚 𝐵) → 𝐺:𝐵𝐶)
3 id 22 . . 3 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
4 fresaun 5988 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
51, 2, 3, 4syl3an 1360 . 2 ((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
6 elmapex 7764 . . . . 5 (𝐹 ∈ (𝐶𝑚 𝐴) → (𝐶 ∈ V ∧ 𝐴 ∈ V))
76simpld 474 . . . 4 (𝐹 ∈ (𝐶𝑚 𝐴) → 𝐶 ∈ V)
873ad2ant1 1075 . . 3 ((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → 𝐶 ∈ V)
96simprd 478 . . . . 5 (𝐹 ∈ (𝐶𝑚 𝐴) → 𝐴 ∈ V)
10 elmapex 7764 . . . . . 6 (𝐺 ∈ (𝐶𝑚 𝐵) → (𝐶 ∈ V ∧ 𝐵 ∈ V))
1110simprd 478 . . . . 5 (𝐺 ∈ (𝐶𝑚 𝐵) → 𝐵 ∈ V)
12 unexg 6857 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
139, 11, 12syl2an 493 . . . 4 ((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵)) → (𝐴𝐵) ∈ V)
14133adant3 1074 . . 3 ((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐴𝐵) ∈ V)
158, 14elmapd 7758 . 2 ((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ∈ (𝐶𝑚 (𝐴𝐵)) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶))
165, 15mpbird 246 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 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ↾ cres 5040  ⟶wf 5800  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by:  diophin  36354  eldioph4b  36393  diophren  36395
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