Theorem elmapresaun 8909

Description: fresaun 6773 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)

Assertion

Ref	Expression
elmapresaun	⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)))

Proof of Theorem elmapresaun

Step	Hyp	Ref	Expression
1		elmapi 8878	. . 3 ⊢ (𝐹 ∈ (𝐶 ↑m 𝐴) → 𝐹:𝐴⟶𝐶)
2		elmapi 8878	. . 3 ⊢ (𝐺 ∈ (𝐶 ↑m 𝐵) → 𝐺:𝐵⟶𝐶)
3		id 22	. . 3 ⊢ ((𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)) → (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵)))
4		fresaun 6773	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)
5	1, 2, 3, 4	syl3an 1157	. 2 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)
6		elmapex 8877	. . . . 5 ⊢ (𝐹 ∈ (𝐶 ↑m 𝐴) → (𝐶 ∈ V ∧ 𝐴 ∈ V))
7	6	simpld 493	. . . 4 ⊢ (𝐹 ∈ (𝐶 ↑m 𝐴) → 𝐶 ∈ V)
8	7	3ad2ant1 1130	. . 3 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → 𝐶 ∈ V)
9	6	simprd 494	. . . . 5 ⊢ (𝐹 ∈ (𝐶 ↑m 𝐴) → 𝐴 ∈ V)
10		elmapex 8877	. . . . . 6 ⊢ (𝐺 ∈ (𝐶 ↑m 𝐵) → (𝐶 ∈ V ∧ 𝐵 ∈ V))
11	10	simprd 494	. . . . 5 ⊢ (𝐺 ∈ (𝐶 ↑m 𝐵) → 𝐵 ∈ V)
12		unexg 7757	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
13	9, 11, 12	syl2an 594	. . . 4 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵)) → (𝐴 ∪ 𝐵) ∈ V)
14	13	3adant3 1129	. . 3 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐴 ∪ 𝐵) ∈ V)
15	8, 14	elmapd 8869	. 2 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ↔ (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶))
16	5, 15	mpbird 256	1 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  Vcvv 3462   ∪ cun 3945   ∩ cin 3946   ↾ cres 5684  ⟶wf 6550  (class class class)co 7424   ↑_m cmap 8855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-map 8857

This theorem is referenced by:  satfv1lem  35190  diophin  42429  eldioph4b  42468  diophren  42470