Theorem brimageg 32997

Description: Closed form of brimage 32996. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
brimageg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)))

Proof of Theorem brimageg

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

Step	Hyp	Ref	Expression
1		breq1 4965	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥Image𝑅𝑦 ↔ 𝐴Image𝑅𝑦))
2		imaeq2 5802	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑅 “ 𝑥) = (𝑅 “ 𝐴))
3	2	eqeq2d 2805	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 = (𝑅 “ 𝑥) ↔ 𝑦 = (𝑅 “ 𝐴)))
4	1, 3	bibi12d 347	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝑥)) ↔ (𝐴Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝐴))))
5		breq2 4966	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴Image𝑅𝑦 ↔ 𝐴Image𝑅𝐵))
6		eqeq1 2799	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝑅 “ 𝐴) ↔ 𝐵 = (𝑅 “ 𝐴)))
7	5, 6	bibi12d 347	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝐴)) ↔ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴))))
8		vex 3440	. . 3 ⊢ 𝑥 ∈ V
9		vex 3440	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	brimage 32996	. 2 ⊢ (𝑥Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝑥))
11	4, 7, 10	vtocl2g 3514	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081   class class class wbr 4962   “ cima 5446  Imagecimage 32910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-symdif 4139  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-eprel 5353  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fo 6231  df-fv 6233  df-1st 7545  df-2nd 7546  df-txp 32924  df-image 32934

This theorem is referenced by:  fnimage  32999