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Theorem fun11uni 7909

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 3066	. . 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 6591	. . 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 3066	. . 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 7908	. . 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 3044 ⊆ wss 3914 ∪ cuni 4871 ^◡ccnv 5637 Fun wfun 6505

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-fun 6513

This theorem is referenced by: f1iun 7922

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