Theorem funimage 36161

Description: Image𝐴 is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
funimage	⊢ Fun Image𝐴

Proof of Theorem funimage

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

Step	Hyp	Ref	Expression
1		difss 4073	. . . 4 ⊢ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ⊆ (V × V)
2		df-rel 5632	. . . 4 ⊢ (Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ⊆ (V × V))
3	1, 2	mpbir 232	. . 3 ⊢ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))
4		df-image 36097	. . . 4 ⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))
5	4	releqi 5728	. . 3 ⊢ (Rel Image𝐴 ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))))
6	3, 5	mpbir 232	. 2 ⊢ Rel Image𝐴
7		vex 3436	. . . . . 6 ⊢ 𝑥 ∈ V
8		vex 3436	. . . . . 6 ⊢ 𝑦 ∈ V
9	7, 8	brimage 36159	. . . . 5 ⊢ (𝑥Image𝐴𝑦 ↔ 𝑦 = (𝐴 “ 𝑥))
10		vex 3436	. . . . . 6 ⊢ 𝑧 ∈ V
11	7, 10	brimage 36159	. . . . 5 ⊢ (𝑥Image𝐴𝑧 ↔ 𝑧 = (𝐴 “ 𝑥))
12		eqtr3 2762	. . . . 5 ⊢ ((𝑦 = (𝐴 “ 𝑥) ∧ 𝑧 = (𝐴 “ 𝑥)) → 𝑦 = 𝑧)
13	9, 11, 12	syl2anb 604	. . . 4 ⊢ ((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
14	13	gen2 1803	. . 3 ⊢ ∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
15	14	ax-gen 1802	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
16		dffun2 6502	. 2 ⊢ (Fun Image𝐴 ↔ (Rel Image𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧)))
17	6, 15, 16	mpbir2an 717	1 ⊢ Fun Image𝐴

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   = wceq 1547  Vcvv 3432   ∖ cdif 3887   ⊆ wss 3890   △ csymdif 4187   class class class wbr 5079   E cep 5524   × cxp 5623  ^◡ccnv 5624  ran crn 5626   “ cima 5628   ∘ ccom 5629  Rel wrel 5630  Fun wfun 6486   ⊗ ctxp 36063  Imagecimage 36073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-symdif 4188  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7938  df-2nd 7939  df-txp 36087  df-image 36097

This theorem is referenced by:  fnimage  36162  imageval  36163  imagesset  36188