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Theorem elmapresaun 37860
Description: fresaun 6216 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 8035 . . 3 (𝐹 ∈ (𝐶𝑚 𝐴) → 𝐹:𝐴𝐶)
2 elmapi 8035 . . 3 (𝐺 ∈ (𝐶𝑚 𝐵) → 𝐺:𝐵𝐶)
3 id 22 . . 3 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
4 fresaun 6216 . . 3 ((𝐹:𝐴𝐶𝐺:𝐵𝐶 ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
51, 2, 3, 4syl3an 1163 . 2 ((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
6 elmapex 8034 . . . . 5 (𝐹 ∈ (𝐶𝑚 𝐴) → (𝐶 ∈ V ∧ 𝐴 ∈ V))
76simpld 482 . . . 4 (𝐹 ∈ (𝐶𝑚 𝐴) → 𝐶 ∈ V)
873ad2ant1 1127 . . 3 ((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → 𝐶 ∈ V)
96simprd 483 . . . . 5 (𝐹 ∈ (𝐶𝑚 𝐴) → 𝐴 ∈ V)
10 elmapex 8034 . . . . . 6 (𝐺 ∈ (𝐶𝑚 𝐵) → (𝐶 ∈ V ∧ 𝐵 ∈ V))
1110simprd 483 . . . . 5 (𝐺 ∈ (𝐶𝑚 𝐵) → 𝐵 ∈ V)
12 unexg 7110 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
139, 11, 12syl2an 583 . . . 4 ((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵)) → (𝐴𝐵) ∈ V)
14133adant3 1126 . . 3 ((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → (𝐴𝐵) ∈ V)
158, 14elmapd 8027 . 2 ((𝐹 ∈ (𝐶𝑚 𝐴) ∧ 𝐺 ∈ (𝐶𝑚 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ∈ (𝐶𝑚 (𝐴𝐵)) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶))
165, 15mpbird 247 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 1071   = wceq 1631  wcel 2145  Vcvv 3351  cun 3721  cin 3722  cres 5252  wf 6026  (class class class)co 6796  𝑚 cmap 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-map 8015
This theorem is referenced by:  diophin  37862  eldioph4b  37901  diophren  37903
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