Theorem infcllem 9415

Description: Lemma for infcl 9416, inflb 9417, infglb 9418, 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 3448	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vex 3448	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	2, 3	brcnv 5836	. . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
5	4	bicomi 224	. . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ 𝑥◡𝑅𝑦)
6	5	notbii 320	. . . . 5 ⊢ (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑥◡𝑅𝑦)
7	6	ralbii 3075	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦)
8	3, 2	brcnv 5836	. . . . . . 7 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
9	8	bicomi 224	. . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 𝑦◡𝑅𝑥)
10		vex 3448	. . . . . . . . 9 ⊢ 𝑧 ∈ V
11	3, 10	brcnv 5836	. . . . . . . 8 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
12	11	bicomi 224	. . . . . . 7 ⊢ (𝑧𝑅𝑦 ↔ 𝑦◡𝑅𝑧)
13	12	rexbii 3076	. . . . . 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 3075	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))
16	7, 15	anbi12i 628	. . 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 3076	. 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 218	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wral 3044  ∃wrex 3053   class class class wbr 5102   Or wor 5538  ^◡ccnv 5630

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-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-cnv 5639

This theorem is referenced by:  infcl  9416  inflb  9417  infglb  9418  infglbb  9419  infiso  9437