fun11uni - Metamath Proof Explorer

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Theorem fun11uni 7928

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 487	. . . . 5 ⊢ ((Fun 𝑓 ∧ Fun ^◡𝑓) → Fun 𝑓)
2	1	anim1i 626	. . . 4 ⊢ (((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)))
3	2	ralimi 3101	. . 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 6611	. . 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 489	. . . . 5 ⊢ ((Fun 𝑓 ∧ Fun ^◡𝑓) → Fun ^◡𝑓)
7	6	anim1i 626	. . . 4 ⊢ (((Fun 𝑓 ∧ Fun ^◡𝑓) ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → (Fun ^◡𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)))
8	7	ralimi 3101	. . 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 7927	. . 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 520	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 400 ∨ wo 860 ∀wral 3078 ⊆ wss 3904 ∪ cuni 4871 ^◡ccnv 5659 Fun wfun 6530

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-pow 5335 ax-pr 5403 ax-un 7734

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-sb 2096 df-clab 2741 df-cleq 2754 df-clel 2837 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-id 5555 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-fun 6538

This theorem is used by: f1iun 7939

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