Theorem fnasrn 5417

Description: A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.)

Assertion

Ref	Expression
fnasrn	⊢ (F Fn A → F = ran {⟨x, y⟩ ∣ (x ∈ A ∧ y = ⟨x, (F ‘x)⟩)})

Distinct variable groups:   x,y,A   x,F,y

Proof of Theorem fnasrn

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

Step	Hyp	Ref	Expression
1		fndm 5182	. . . . . 6 ⊢ (F Fn A → dom F = A)
2		opeldm 4910	. . . . . . 7 ⊢ (⟨z, w⟩ ∈ F → z ∈ dom F)
3		eleq2 2414	. . . . . . 7 ⊢ (dom F = A → (z ∈ dom F ↔ z ∈ A))
4	2, 3	syl5ib 210	. . . . . 6 ⊢ (dom F = A → (⟨z, w⟩ ∈ F → z ∈ A))
5	1, 4	syl 15	. . . . 5 ⊢ (F Fn A → (⟨z, w⟩ ∈ F → z ∈ A))
6		eleq1 2413	. . . . . . . . 9 ⊢ (x = z → (x ∈ A ↔ z ∈ A))
7	6	biimpcd 215	. . . . . . . 8 ⊢ (x ∈ A → (x = z → z ∈ A))
8	7	adantrd 454	. . . . . . 7 ⊢ (x ∈ A → ((x = z ∧ (F ‘x) = w) → z ∈ A))
9	8	rexlimiv 2732	. . . . . 6 ⊢ (∃x ∈ A (x = z ∧ (F ‘x) = w) → z ∈ A)
10	9	a1i 10	. . . . 5 ⊢ (F Fn A → (∃x ∈ A (x = z ∧ (F ‘x) = w) → z ∈ A))
11		fveq2 5328	. . . . . . . . . 10 ⊢ (x = z → (F ‘x) = (F ‘z))
12	11	eqeq1d 2361	. . . . . . . . 9 ⊢ (x = z → ((F ‘x) = w ↔ (F ‘z) = w))
13	12	ceqsrexv 2972	. . . . . . . 8 ⊢ (z ∈ A → (∃x ∈ A (x = z ∧ (F ‘x) = w) ↔ (F ‘z) = w))
14	13	adantl 452	. . . . . . 7 ⊢ ((F Fn A ∧ z ∈ A) → (∃x ∈ A (x = z ∧ (F ‘x) = w) ↔ (F ‘z) = w))
15		fnopfvb 5359	. . . . . . 7 ⊢ ((F Fn A ∧ z ∈ A) → ((F ‘z) = w ↔ ⟨z, w⟩ ∈ F))
16	14, 15	bitr2d 245	. . . . . 6 ⊢ ((F Fn A ∧ z ∈ A) → (⟨z, w⟩ ∈ F ↔ ∃x ∈ A (x = z ∧ (F ‘x) = w)))
17	16	ex 423	. . . . 5 ⊢ (F Fn A → (z ∈ A → (⟨z, w⟩ ∈ F ↔ ∃x ∈ A (x = z ∧ (F ‘x) = w))))
18	5, 10, 17	pm5.21ndd 343	. . . 4 ⊢ (F Fn A → (⟨z, w⟩ ∈ F ↔ ∃x ∈ A (x = z ∧ (F ‘x) = w)))
19		vex 2862	. . . . . 6 ⊢ z ∈ V
20		vex 2862	. . . . . 6 ⊢ w ∈ V
21	19, 20	opex 4588	. . . . 5 ⊢ ⟨z, w⟩ ∈ V
22		eqeq1 2359	. . . . . . 7 ⊢ (y = ⟨z, w⟩ → (y = ⟨x, (F ‘x)⟩ ↔ ⟨z, w⟩ = ⟨x, (F ‘x)⟩))
23		eqcom 2355	. . . . . . . 8 ⊢ (⟨z, w⟩ = ⟨x, (F ‘x)⟩ ↔ ⟨x, (F ‘x)⟩ = ⟨z, w⟩)
24		opth 4602	. . . . . . . 8 ⊢ (⟨x, (F ‘x)⟩ = ⟨z, w⟩ ↔ (x = z ∧ (F ‘x) = w))
25	23, 24	bitri 240	. . . . . . 7 ⊢ (⟨z, w⟩ = ⟨x, (F ‘x)⟩ ↔ (x = z ∧ (F ‘x) = w))
26	22, 25	syl6bb 252	. . . . . 6 ⊢ (y = ⟨z, w⟩ → (y = ⟨x, (F ‘x)⟩ ↔ (x = z ∧ (F ‘x) = w)))
27	26	rexbidv 2635	. . . . 5 ⊢ (y = ⟨z, w⟩ → (∃x ∈ A y = ⟨x, (F ‘x)⟩ ↔ ∃x ∈ A (x = z ∧ (F ‘x) = w)))
28	21, 27	elab 2985	. . . 4 ⊢ (⟨z, w⟩ ∈ {y ∣ ∃x ∈ A y = ⟨x, (F ‘x)⟩} ↔ ∃x ∈ A (x = z ∧ (F ‘x) = w))
29	18, 28	syl6bbr 254	. . 3 ⊢ (F Fn A → (⟨z, w⟩ ∈ F ↔ ⟨z, w⟩ ∈ {y ∣ ∃x ∈ A y = ⟨x, (F ‘x)⟩}))
30	29	eqrelrdv 4852	. 2 ⊢ (F Fn A → F = {y ∣ ∃x ∈ A y = ⟨x, (F ‘x)⟩})
31		rnopab2 4968	. 2 ⊢ ran {⟨x, y⟩ ∣ (x ∈ A ∧ y = ⟨x, (F ‘x)⟩)} = {y ∣ ∃x ∈ A y = ⟨x, (F ‘x)⟩}
32	30, 31	syl6eqr 2403	1 ⊢ (F Fn A → F = ran {⟨x, y⟩ ∣ (x ∈ A ∧ y = ⟨x, (F ‘x)⟩)})

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  ⟨cop 4561  {copab 4622  dom cdm 4772  ran crn 4773   Fn wfn 4776   ‘cfv 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795

This theorem is referenced by: (None)