Theorem elmapresaun 37334

Description: fresaun 6075 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 7879	. . 3 ⊢ (𝐹 ∈ (𝐶 ↑𝑚 𝐴) → 𝐹:𝐴⟶𝐶)
2		elmapi 7879	. . 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 6075	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)
5	1, 2, 3, 4	syl3an 1368	. 2 ⊢ ((𝐹 ∈ (𝐶 ↑𝑚 𝐴) ∧ 𝐺 ∈ (𝐶 ↑𝑚 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)
6		elmapex 7878	. . . . 5 ⊢ (𝐹 ∈ (𝐶 ↑𝑚 𝐴) → (𝐶 ∈ V ∧ 𝐴 ∈ V))
7	6	simpld 475	. . . 4 ⊢ (𝐹 ∈ (𝐶 ↑𝑚 𝐴) → 𝐶 ∈ V)
8	7	3ad2ant1 1082	. . 3 ⊢ ((𝐹 ∈ (𝐶 ↑𝑚 𝐴) ∧ 𝐺 ∈ (𝐶 ↑𝑚 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → 𝐶 ∈ V)
9	6	simprd 479	. . . . 5 ⊢ (𝐹 ∈ (𝐶 ↑𝑚 𝐴) → 𝐴 ∈ V)
10		elmapex 7878	. . . . . 6 ⊢ (𝐺 ∈ (𝐶 ↑𝑚 𝐵) → (𝐶 ∈ V ∧ 𝐵 ∈ V))
11	10	simprd 479	. . . . 5 ⊢ (𝐺 ∈ (𝐶 ↑𝑚 𝐵) → 𝐵 ∈ V)
12		unexg 6959	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
13	9, 11, 12	syl2an 494	. . . 4 ⊢ ((𝐹 ∈ (𝐶 ↑𝑚 𝐴) ∧ 𝐺 ∈ (𝐶 ↑𝑚 𝐵)) → (𝐴 ∪ 𝐵) ∈ V)
14	13	3adant3 1081	. . 3 ⊢ ((𝐹 ∈ (𝐶 ↑𝑚 𝐴) ∧ 𝐺 ∈ (𝐶 ↑𝑚 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐴 ∪ 𝐵) ∈ V)
15	8, 14	elmapd 7871	. 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 247	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 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ↾ cres 5116  ⟶wf 5884  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  diophin  37336  eldioph4b  37375  diophren  37377