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Theorem fun11uni 7937

Description: The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)

**Assertion**
Ref	Expression
fun11uni	⊢ (∀𝑓 ∈ 𝐴 ((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun ∪ 𝐴 ∧ Fun ^◡∪ 𝐴))

Distinct variable group: 𝑓,𝑔,𝐴

**Proof of Theorem fun11uni**
Step	Hyp	Ref	Expression
1		simpl 482	. . . . 5 ⊢ ((Fun 𝑓 ∧ Fun ^◡𝑓) → Fun 𝑓)
2	1	anim1i 615	. . . 4 ⊢ (((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)))
3	2	ralimi 3072	. . 3 ⊢ (∀𝑓 ∈ 𝐴 ((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)))
4		fununi 6621	. . 3 ⊢ (∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ∪ 𝐴)
5	3, 4	syl 17	. 2 ⊢ (∀𝑓 ∈ 𝐴 ((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ∪ 𝐴)
6		simpr 484	. . . . 5 ⊢ ((Fun 𝑓 ∧ Fun ^◡𝑓) → Fun ^◡𝑓)
7	6	anim1i 615	. . . 4 ⊢ (((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun ^◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)))
8	7	ralimi 3072	. . 3 ⊢ (∀𝑓 ∈ 𝐴 ((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → ∀𝑓 ∈ 𝐴 (Fun ^◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)))
9		funcnvuni 7936	. . 3 ⊢ (∀𝑓 ∈ 𝐴 (Fun ^◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ^◡∪ 𝐴)
10	8, 9	syl 17	. 2 ⊢ (∀𝑓 ∈ 𝐴 ((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ^◡∪ 𝐴)
11	5, 10	jca 511	1 ⊢ (∀𝑓 ∈ 𝐴 ((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun ∪ 𝐴 ∧ Fun ^◡∪ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∀wral 3050 ⊆ wss 3931 ∪ cuni 4887 ^◡ccnv 5664 Fun wfun 6535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ral 3051 df-rex 3060 df-rab 3420 df-v 3465 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-fun 6543

This theorem is referenced by: f1iun 7950

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