Theorem f1oun 5591

Description: The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)

Assertion

Ref	Expression
f1oun	⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷))

Proof of Theorem f1oun

Step	Hyp	Ref	Expression
1		dff1o4 5579	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
2		dff1o4 5579	. . . 4 ⊢ (𝐺:𝐶–1-1-onto→𝐷 ↔ (𝐺 Fn 𝐶 ∧ ◡𝐺 Fn 𝐷))
3		fnun 5428	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶))
4	3	ex 115	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) → ((𝐴 ∩ 𝐶) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶)))
5		fnun 5428	. . . . . . . 8 ⊢ (((◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (◡𝐹 ∪ ◡𝐺) Fn (𝐵 ∪ 𝐷))
6		cnvun 5133	. . . . . . . . 9 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
7	6	fneq1i 5414	. . . . . . . 8 ⊢ (◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷) ↔ (◡𝐹 ∪ ◡𝐺) Fn (𝐵 ∪ 𝐷))
8	5, 7	sylibr 134	. . . . . . 7 ⊢ (((◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷))
9	8	ex 115	. . . . . 6 ⊢ ((◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷)))
10	4, 9	im2anan9 600	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷)) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷))))
11	10	an4s 590	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵) ∧ (𝐺 Fn 𝐶 ∧ ◡𝐺 Fn 𝐷)) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷))))
12	1, 2, 11	syl2anb 291	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷))))
13		dff1o4 5579	. . 3 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷)))
14	12, 13	imbitrrdi 162	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷)))
15	14	imp 124	1 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∪ cun 3195   ∩ cin 3196  ∅c0 3491  ^◡ccnv 4717   Fn wfn 5312  –1-1-onto→wf1o 5316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324

This theorem is referenced by:  f1oprg  5616  unen  6967  zfz1isolem1  11057  ennnfonelemhf1o  12979