Theorem fliftcnv 6704

Description: Converse of the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
flift.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
flift.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)

Assertion

Ref	Expression
fliftcnv	⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))

Distinct variable groups:   𝑥,𝑅   𝜑,𝑥   𝑥,𝑋   𝑥,𝑆

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftcnv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2771	. . . . 5 ⊢ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩) = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)
2		flift.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
3		flift.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
4	1, 2, 3	fliftrel 6701	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅))
5		relxp 5266	. . . 4 ⊢ Rel (𝑆 × 𝑅)
6		relss 5346	. . . 4 ⊢ (ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩) ⊆ (𝑆 × 𝑅) → (Rel (𝑆 × 𝑅) → Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)))
7	4, 5, 6	mpisyl 21	. . 3 ⊢ (𝜑 → Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
8		relcnv 5644	. . 3 ⊢ Rel ◡𝐹
9	7, 8	jctil 509	. 2 ⊢ (𝜑 → (Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)))
10		flift.1	. . . . . . 7 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
11	10, 3, 2	fliftel 6702	. . . . . 6 ⊢ (𝜑 → (𝑧𝐹𝑦 ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵)))
12		vex 3354	. . . . . . 7 ⊢ 𝑦 ∈ V
13		vex 3354	. . . . . . 7 ⊢ 𝑧 ∈ V
14	12, 13	brcnv 5443	. . . . . 6 ⊢ (𝑦◡𝐹𝑧 ↔ 𝑧𝐹𝑦)
15		ancom 452	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))
16	15	rexbii 3189	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴) ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑦 = 𝐵))
17	11, 14, 16	3bitr4g 303	. . . . 5 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴)))
18	1, 2, 3	fliftel 6702	. . . . 5 ⊢ (𝜑 → (𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ∃𝑥 ∈ 𝑋 (𝑦 = 𝐵 ∧ 𝑧 = 𝐴)))
19	17, 18	bitr4d 271	. . . 4 ⊢ (𝜑 → (𝑦◡𝐹𝑧 ↔ 𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧))
20		df-br 4787	. . . 4 ⊢ (𝑦◡𝐹𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ◡𝐹)
21		df-br 4787	. . . 4 ⊢ (𝑦ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
22	19, 20, 21	3bitr3g 302	. . 3 ⊢ (𝜑 → (⟨𝑦, 𝑧⟩ ∈ ◡𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)))
23	22	eqrelrdv2 5359	. 2 ⊢ (((Rel ◡𝐹 ∧ Rel ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)) ∧ 𝜑) → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))
24	9, 23	mpancom 668	1 ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∃wrex 3062   ⊆ wss 3723  ⟨cop 4322   class class class wbr 4786   ↦ cmpt 4863   × cxp 5247  ^◡ccnv 5248  ran crn 5250  Rel wrel 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039

This theorem is referenced by:  pi1xfrcnvlem  23075