Theorem funimage 34623

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 4111	. . . 4 ⊢ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ⊆ (V × V)
2		df-rel 5660	. . . 4 ⊢ (Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))) ⊆ (V × V))
3	1, 2	mpbir 230	. . 3 ⊢ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))
4		df-image 34559	. . . 4 ⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V)))
5	4	releqi 5753	. . 3 ⊢ (Rel Image𝐴 ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝐴) ⊗ V))))
6	3, 5	mpbir 230	. 2 ⊢ Rel Image𝐴
7		vex 3463	. . . . . 6 ⊢ 𝑥 ∈ V
8		vex 3463	. . . . . 6 ⊢ 𝑦 ∈ V
9	7, 8	brimage 34621	. . . . 5 ⊢ (𝑥Image𝐴𝑦 ↔ 𝑦 = (𝐴 “ 𝑥))
10		vex 3463	. . . . . 6 ⊢ 𝑧 ∈ V
11	7, 10	brimage 34621	. . . . 5 ⊢ (𝑥Image𝐴𝑧 ↔ 𝑧 = (𝐴 “ 𝑥))
12		eqtr3 2757	. . . . 5 ⊢ ((𝑦 = (𝐴 “ 𝑥) ∧ 𝑧 = (𝐴 “ 𝑥)) → 𝑦 = 𝑧)
13	9, 11, 12	syl2anb 598	. . . 4 ⊢ ((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
14	13	gen2 1798	. . 3 ⊢ ∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
15	14	ax-gen 1797	. 2 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
16		dffun2 6526	. 2 ⊢ (Fun Image𝐴 ↔ (Rel Image𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥Image𝐴𝑦 ∧ 𝑥Image𝐴𝑧) → 𝑦 = 𝑧)))
17	6, 15, 16	mpbir2an 709	1 ⊢ Fun Image𝐴

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1539   = wceq 1541  Vcvv 3459   ∖ cdif 3925   ⊆ wss 3928   △ csymdif 4221   class class class wbr 5125   E cep 5556   × cxp 5651  ^◡ccnv 5652  ran crn 5654   “ cima 5656   ∘ ccom 5657  Rel wrel 5658  Fun wfun 6510   ⊗ ctxp 34525  Imagecimage 34535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404  ax-un 7692

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-symdif 4222  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-eprel 5557  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-fo 6522  df-fv 6524  df-1st 7941  df-2nd 7942  df-txp 34549  df-image 34559

This theorem is referenced by:  fnimage  34624  imageval  34625  imagesset  34648