Theorem dffun6f 5123

Description: Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by Scott Fenton, 16-Apr-2021.)

Hypotheses

Ref	Expression
dffun6f.1	⊢ ℲxA
dffun6f.2	⊢ ℲyA

Assertion

Ref	Expression
dffun6f	⊢ (Fun A ↔ ∀x∃*y xAy)

Distinct variable group:   x,y

Allowed substitution hints:   A(x,y)

Proof of Theorem dffun6f

Dummy variables w v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun3 5120	. 2 ⊢ (Fun A ↔ ∀w∃u∀v(wAv → v = u))
2		nfcv 2489	. . . . . 6 ⊢ Ⅎyw
3		dffun6f.2	. . . . . 6 ⊢ ℲyA
4		nfcv 2489	. . . . . 6 ⊢ Ⅎyv
5	2, 3, 4	nfbr 4683	. . . . 5 ⊢ Ⅎy wAv
6		nfv 1619	. . . . 5 ⊢ Ⅎv wAy
7		breq2 4643	. . . . 5 ⊢ (v = y → (wAv ↔ wAy))
8	5, 6, 7	cbvmo 2241	. . . 4 ⊢ (∃*v wAv ↔ ∃*y wAy)
9	8	albii 1566	. . 3 ⊢ (∀w∃*v wAv ↔ ∀w∃*y wAy)
10		nfv 1619	. . . . 5 ⊢ Ⅎu wAv
11	10	mo2 2233	. . . 4 ⊢ (∃*v wAv ↔ ∃u∀v(wAv → v = u))
12	11	albii 1566	. . 3 ⊢ (∀w∃*v wAv ↔ ∀w∃u∀v(wAv → v = u))
13		nfcv 2489	. . . . . 6 ⊢ Ⅎxw
14		dffun6f.1	. . . . . 6 ⊢ ℲxA
15		nfcv 2489	. . . . . 6 ⊢ Ⅎxy
16	13, 14, 15	nfbr 4683	. . . . 5 ⊢ Ⅎx wAy
17	16	nfmo 2221	. . . 4 ⊢ Ⅎx∃*y wAy
18		nfv 1619	. . . 4 ⊢ Ⅎw∃*y xAy
19		breq1 4642	. . . . 5 ⊢ (w = x → (wAy ↔ xAy))
20	19	mobidv 2239	. . . 4 ⊢ (w = x → (∃*y wAy ↔ ∃*y xAy))
21	17, 18, 20	cbval 1984	. . 3 ⊢ (∀w∃*y wAy ↔ ∀x∃*y xAy)
22	9, 12, 21	3bitr3ri 267	. 2 ⊢ (∀x∃*y xAy ↔ ∀w∃u∀v(wAv → v = u))
23	1, 22	bitr4i 243	1 ⊢ (Fun A ↔ ∀x∃*y xAy)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  ∃*wmo 2205  Ⅎwnfc 2476   class class class wbr 4639  Fun wfun 4775

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-id 4767  df-cnv 4785  df-fun 4789

This theorem is referenced by:  dffun6  5124  funopab  5139