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Theorem infcllem 9246

Description: Lemma for infcl 9247, inflb 9248, infglb 9249, 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 3436	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vex 3436	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	2, 3	brcnv 5791	. . . . . . 7 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
5	4	bicomi 223	. . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ 𝑥^◡𝑅𝑦)
6	5	notbii 320	. . . . 5 ⊢ (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑥^◡𝑅𝑦)
7	6	ralbii 3092	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑥^◡𝑅𝑦)
8	3, 2	brcnv 5791	. . . . . . 7 ⊢ (𝑦^◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
9	8	bicomi 223	. . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 𝑦^◡𝑅𝑥)
10		vex 3436	. . . . . . . . 9 ⊢ 𝑧 ∈ V
11	3, 10	brcnv 5791	. . . . . . . 8 ⊢ (𝑦^◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
12	11	bicomi 223	. . . . . . 7 ⊢ (𝑧𝑅𝑦 ↔ 𝑦^◡𝑅𝑧)
13	12	rexbii 3181	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 𝑧𝑅𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑦^◡𝑅𝑧)
14	9, 13	imbi12i 351	. . . . 5 ⊢ ((𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ↔ (𝑦^◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦^◡𝑅𝑧))
15	14	ralbii 3092	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑦^◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦^◡𝑅𝑧))
16	7, 15	anbi12i 627	. . 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 3181	. 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 396 ∀wral 3064 ∃wrex 3065 class class class wbr 5074 Or wor 5502 ^◡ccnv 5588

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-cnv 5597

This theorem is referenced by: infcl 9247 inflb 9248 infglb 9249 infglbb 9250 infiso 9267

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