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Mirrors > Home > NFE Home > Th. List > dffun7 | GIF version |
Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one." However, dffun8 5134 shows that it doesn't matter which meaning we pick.) (Contributed by set.mm contributors, 4-Nov-2002.) (Revised by Scott Fenton, 16-Apr-2021.) |
Ref | Expression |
---|---|
dffun7 | ⊢ (Fun A ↔ ∀x ∈ dom A∃*y xAy) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moabs 2248 | . . . 4 ⊢ (∃*y xAy ↔ (∃y xAy → ∃*y xAy)) | |
2 | eldm 4898 | . . . . 5 ⊢ (x ∈ dom A ↔ ∃y xAy) | |
3 | 2 | imbi1i 315 | . . . 4 ⊢ ((x ∈ dom A → ∃*y xAy) ↔ (∃y xAy → ∃*y xAy)) |
4 | 1, 3 | bitr4i 243 | . . 3 ⊢ (∃*y xAy ↔ (x ∈ dom A → ∃*y xAy)) |
5 | 4 | albii 1566 | . 2 ⊢ (∀x∃*y xAy ↔ ∀x(x ∈ dom A → ∃*y xAy)) |
6 | dffun6 5124 | . 2 ⊢ (Fun A ↔ ∀x∃*y xAy) | |
7 | df-ral 2619 | . 2 ⊢ (∀x ∈ dom A∃*y xAy ↔ ∀x(x ∈ dom A → ∃*y xAy)) | |
8 | 5, 6, 7 | 3bitr4i 268 | 1 ⊢ (Fun A ↔ ∀x ∈ dom A∃*y xAy) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 ∃wex 1541 ∈ wcel 1710 ∃*wmo 2205 ∀wral 2614 class class class wbr 4639 dom cdm 4772 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-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 |
This theorem is referenced by: dffun8 5134 dffun9 5135 |
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