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Theorem infcllem 9176

Description: Lemma for infcl 9177, inflb 9178, infglb 9179, etc. (Contributed by AV, 3-Sep-2020.)

**Hypotheses**
Ref	Expression
infcl.1	⊢ (𝜑 → 𝑅 Or 𝐴)
infcl.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))

**Assertion**
Ref	Expression
infcllem	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥^◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦^◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦^◡𝑅𝑧)))

Distinct variable groups: 𝑥,𝐴,𝑦,𝑧 𝑥,𝐵,𝑦,𝑧 𝑥,𝑅,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑧)

**Proof of Theorem infcllem**
Step	Hyp	Ref	Expression
1		infcl.2	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
2		vex 3426	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vex 3426	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	2, 3	brcnv 5780	. . . . . . 7 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
5	4	bicomi 223	. . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ 𝑥^◡𝑅𝑦)
6	5	notbii 319	. . . . 5 ⊢ (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑥^◡𝑅𝑦)
7	6	ralbii 3090	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑥^◡𝑅𝑦)
8	3, 2	brcnv 5780	. . . . . . 7 ⊢ (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
9	8	bicomi 223	. . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 𝑦^◡𝑅𝑥)
10		vex 3426	. . . . . . . . 9 ⊢ 𝑧 ∈ V
11	3, 10	brcnv 5780	. . . . . . . 8 ⊢ (𝑦^◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
12	11	bicomi 223	. . . . . . 7 ⊢ (𝑧𝑅𝑦 ↔ 𝑦^◡𝑅𝑧)
13	12	rexbii 3177	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 𝑧𝑅𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑦^◡𝑅𝑧)
14	9, 13	imbi12i 350	. . . . 5 ⊢ ((𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ↔ (𝑦^◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦^◡𝑅𝑧))
15	14	ralbii 3090	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑦^◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦^◡𝑅𝑧))
16	7, 15	anbi12i 626	. . 3 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝑥^◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦^◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦^◡𝑅𝑧)))
17	16	rexbii 3177	. 2 ⊢ (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) ↔ ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥^◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦^◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦^◡𝑅𝑧)))
18	1, 17	sylib 217	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥^◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦^◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦^◡𝑅𝑧)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wral 3063 ∃wrex 3064 class class class wbr 5070 Or wor 5493 ^◡ccnv 5579

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-cnv 5588

This theorem is referenced by: infcl 9177 inflb 9178 infglb 9179 infglbb 9180 infiso 9197

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