Theorem fununfunOLD 6598

Description: Obsolete version of fununfun 6597 as of 24-Oct-2026. (Contributed by AV, 18-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
fununfunOLD	⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺))

Proof of Theorem fununfunOLD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funrel 6565	. . 3 ⊢ (Fun (𝐹 ∪ 𝐺) → Rel (𝐹 ∪ 𝐺))
2		relun 5794	. . 3 ⊢ (Rel (𝐹 ∪ 𝐺) ↔ (Rel 𝐹 ∧ Rel 𝐺))
3	1, 2	sylib 221	. 2 ⊢ (Fun (𝐹 ∪ 𝐺) → (Rel 𝐹 ∧ Rel 𝐺))
4		simpl 488	. . . . 5 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐹)
5		fununmo 6596	. . . . . 6 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦)
6	5	alrimiv 1961	. . . . 5 ⊢ (Fun (𝐹 ∪ 𝐺) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
7	4, 6	anim12i 625	. . . 4 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
8		dffun6 6558	. . . 4 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
9	7, 8	sylibr 237	. . 3 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → Fun 𝐹)
10		simpr 490	. . . . 5 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐺)
11		uncom 4160	. . . . . . . 8 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
12	11	funeqi 6569	. . . . . . 7 ⊢ (Fun (𝐹 ∪ 𝐺) ↔ Fun (𝐺 ∪ 𝐹))
13		fununmo 6596	. . . . . . 7 ⊢ (Fun (𝐺 ∪ 𝐹) → ∃*𝑦 𝑥𝐺𝑦)
14	12, 13	sylbi 220	. . . . . 6 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐺𝑦)
15	14	alrimiv 1961	. . . . 5 ⊢ (Fun (𝐹 ∪ 𝐺) → ∀𝑥∃*𝑦 𝑥𝐺𝑦)
16	10, 15	anim12i 625	. . . 4 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦))
17		dffun6 6558	. . . 4 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦))
18	16, 17	sylibr 237	. . 3 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → Fun 𝐺)
19	9, 18	jca 521	. 2 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Fun 𝐹 ∧ Fun 𝐺))
20	3, 19	mpancom 701	1 ⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567  ∃*wmo 2565   ∪ cun 4152   class class class wbr 5106  Rel wrel 5660  Fun wfun 6541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-fun 6549

This theorem is used by: (None)