Theorem cnvinfex 6898

Description: Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.)

Hypothesis

Ref	Expression
cnvinfex.ex	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))

Assertion

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

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦   𝜑,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝑅(𝑥,𝑦,𝑧)

Proof of Theorem cnvinfex

Step	Hyp	Ref	Expression
1		cnvinfex.ex	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
2		vex 2684	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vex 2684	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	2, 3	brcnv 4717	. . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
5	4	a1i 9	. . . . . 6 ⊢ (𝜑 → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
6	5	notbid 656	. . . . 5 ⊢ (𝜑 → (¬ 𝑥◡𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
7	6	ralbidv 2435	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
8	3, 2	brcnv 4717	. . . . . . 7 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
9	8	a1i 9	. . . . . 6 ⊢ (𝜑 → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
10		vex 2684	. . . . . . . . 9 ⊢ 𝑧 ∈ V
11	3, 10	brcnv 4717	. . . . . . . 8 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
12	11	a1i 9	. . . . . . 7 ⊢ (𝜑 → (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦))
13	12	rexbidv 2436	. . . . . 6 ⊢ (𝜑 → (∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))
14	9, 13	imbi12d 233	. . . . 5 ⊢ (𝜑 → ((𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧) ↔ (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
15	14	ralbidv 2435	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
16	7, 15	anbi12d 464	. . 3 ⊢ (𝜑 → ((∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))
17	16	rexbidv 2436	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)) ↔ ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))
18	1, 17	mpbird 166	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  ∀wral 2414  ∃wrex 2415   class class class wbr 3924  ^◡ccnv 4533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-cnv 4542

This theorem is referenced by:  infvalti  6902  infclti  6903  inflbti  6904  infglbti  6905  infisoti  6912  infrenegsupex  9382  infxrnegsupex  11025