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Theorem ustbas2 24120

Description: Second direction for ustbas 24122. (Contributed by Thierry Arnoux, 16-Nov-2017.)

**Assertion**
Ref	Expression
ustbas2	⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)

Proof of Theorem ustbas2 Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmxpid 5897	. 2 ⊢ dom (𝑋 × 𝑋) = 𝑋
2		ustbasel 24101	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3		elssuni 4904	. . . . 5 ⊢ ((𝑋 × 𝑋) ∈ 𝑈 → (𝑋 × 𝑋) ⊆ ∪ 𝑈)
4	2, 3	syl 17	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ⊆ ∪ 𝑈)
5		elfvex 6899	. . . . . . . . 9 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
6		isust 24098	. . . . . . . . 9 ⊢ (𝑋 ∈ V → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ^◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))))
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ^◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))))
8	7	ibi 267	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ^◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣))))
9	8	simp1d 1142	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
10	9	unissd 4884	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ⊆ ∪ 𝒫 (𝑋 × 𝑋))
11		unipw 5413	. . . . 5 ⊢ ∪ 𝒫 (𝑋 × 𝑋) = (𝑋 × 𝑋)
12	10, 11	sseqtrdi 3990	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ⊆ (𝑋 × 𝑋))
13	4, 12	eqssd 3967	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ∪ 𝑈)
14	13	dmeqd 5872	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → dom (𝑋 × 𝑋) = dom ∪ 𝑈)
15	1, 14	eqtr3id 2779	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ∩ cin 3916 ⊆ wss 3917 𝒫 cpw 4566 ∪ cuni 4874 I cid 5535 × cxp 5639 ^◡ccnv 5640 dom cdm 5641 ↾ cres 5643 ∘ ccom 5645 ‘cfv 6514 UnifOncust 24094

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 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

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-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-res 5653 df-iota 6467 df-fun 6516 df-fv 6522 df-ust 24095

This theorem is referenced by: ustbas 24122 utopval 24127 tuslem 24161 ucnval 24171 iscfilu 24182

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