Theorem elmapresaun 37860

Description: fresaun 6215 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 8031	. . 3 ⊢ (𝐹 ∈ (𝐶 ↑𝑚 𝐴) → 𝐹:𝐴⟶𝐶)
2		elmapi 8031	. . 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 6215	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)
5	1, 2, 3, 4	syl3an 1163	. 2 ⊢ ((𝐹 ∈ (𝐶 ↑𝑚 𝐴) ∧ 𝐺 ∈ (𝐶 ↑𝑚 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)
6		elmapex 8030	. . . . 5 ⊢ (𝐹 ∈ (𝐶 ↑𝑚 𝐴) → (𝐶 ∈ V ∧ 𝐴 ∈ V))
7	6	simpld 482	. . . 4 ⊢ (𝐹 ∈ (𝐶 ↑𝑚 𝐴) → 𝐶 ∈ V)
8	7	3ad2ant1 1127	. . 3 ⊢ ((𝐹 ∈ (𝐶 ↑𝑚 𝐴) ∧ 𝐺 ∈ (𝐶 ↑𝑚 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → 𝐶 ∈ V)
9	6	simprd 483	. . . . 5 ⊢ (𝐹 ∈ (𝐶 ↑𝑚 𝐴) → 𝐴 ∈ V)
10		elmapex 8030	. . . . . 6 ⊢ (𝐺 ∈ (𝐶 ↑𝑚 𝐵) → (𝐶 ∈ V ∧ 𝐵 ∈ V))
11	10	simprd 483	. . . . 5 ⊢ (𝐺 ∈ (𝐶 ↑𝑚 𝐵) → 𝐵 ∈ V)
12		unexg 7106	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
13	9, 11, 12	syl2an 583	. . . 4 ⊢ ((𝐹 ∈ (𝐶 ↑𝑚 𝐴) ∧ 𝐺 ∈ (𝐶 ↑𝑚 𝐵)) → (𝐴 ∪ 𝐵) ∈ V)
14	13	3adant3 1126	. . 3 ⊢ ((𝐹 ∈ (𝐶 ↑𝑚 𝐴) ∧ 𝐺 ∈ (𝐶 ↑𝑚 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐴 ∪ 𝐵) ∈ V)
15	8, 14	elmapd 8023	. 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 1071   = wceq 1631   ∈ wcel 2145  Vcvv 3351   ∪ cun 3721   ∩ cin 3722   ↾ cres 5251  ⟶wf 6027  (class class class)co 6793   ↑_𝑚 cmap 8009

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 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

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 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011

This theorem is referenced by:  diophin  37862  eldioph4b  37901  diophren  37903