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Theorem elmapresaun 8467
 Description: fresaun 6538 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 8443 . . 3 (𝐹 ∈ (𝐶m 𝐴) → 𝐹:𝐴𝐶)
2 elmapi 8443 . . 3 (𝐺 ∈ (𝐶m 𝐵) → 𝐺:𝐵𝐶)
3 id 22 . . 3 ((𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)) → (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵)))
4 fresaun 6538 . . 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 8442 . . . . 5 (𝐹 ∈ (𝐶m 𝐴) → (𝐶 ∈ V ∧ 𝐴 ∈ V))
76simpld 498 . . . 4 (𝐹 ∈ (𝐶m 𝐴) → 𝐶 ∈ V)
873ad2ant1 1130 . . 3 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → 𝐶 ∈ V)
96simprd 499 . . . . 5 (𝐹 ∈ (𝐶m 𝐴) → 𝐴 ∈ V)
10 elmapex 8442 . . . . . 6 (𝐺 ∈ (𝐶m 𝐵) → (𝐶 ∈ V ∧ 𝐵 ∈ V))
1110simprd 499 . . . . 5 (𝐺 ∈ (𝐶m 𝐵) → 𝐵 ∈ V)
12 unexg 7475 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
139, 11, 12syl2an 598 . . . 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 8435 . 2 ((𝐹 ∈ (𝐶m 𝐴) ∧ 𝐺 ∈ (𝐶m 𝐵) ∧ (𝐹 ↾ (𝐴𝐵)) = (𝐺 ↾ (𝐴𝐵))) → ((𝐹𝐺) ∈ (𝐶m (𝐴𝐵)) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶))
165, 15mpbird 260 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 1538   ∈ wcel 2111  Vcvv 3409   ∪ cun 3858   ∩ cin 3859   ↾ cres 5529  ⟶wf 6335  (class class class)co 7155   ↑m cmap 8421 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7698  df-2nd 7699  df-map 8423 This theorem is referenced by:  satfv1lem  32844  diophin  40114  eldioph4b  40153  diophren  40155
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