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Theorem fun11uni 7939

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 488	. . . . 5 ⊢ ((Fun 𝑓 ∧ Fun ^◡𝑓) → Fun 𝑓)
2	1	anim1i 627	. . . 4 ⊢ (((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)))
3	2	ralimi 3105	. . 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 6618	. . 3 ⊢ (∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ∪ 𝐴)
5	3, 4	syl 18	. 2 ⊢ (∀𝑓 ∈ 𝐴 ((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ∪ 𝐴)
6		simpr 490	. . . . 5 ⊢ ((Fun 𝑓 ∧ Fun ^◡𝑓) → Fun ^◡𝑓)
7	6	anim1i 627	. . . 4 ⊢ (((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun ^◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)))
8	7	ralimi 3105	. . 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 7938	. . 3 ⊢ (∀𝑓 ∈ 𝐴 (Fun ^◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ^◡∪ 𝐴)
10	8, 9	syl 18	. 2 ⊢ (∀𝑓 ∈ 𝐴 ((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ^◡∪ 𝐴)
11	5, 10	jca 521	1 ⊢ (∀𝑓 ∈ 𝐴 ((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun ∪ 𝐴 ∧ Fun ^◡∪ 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 ∀wral 3082 ⊆ wss 3908 ∪ cuni 4877 ^◡ccnv 5665 Fun wfun 6537

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-pow 5341 ax-pr 5409 ax-un 7745

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2745 df-cleq 2758 df-clel 2841 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-fun 6545

This theorem is used by: f1iun 7950

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