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Mirrors > Home > NFE Home > Th. List > nffo | GIF version |
Description: Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.) |
Ref | Expression |
---|---|
nffo.1 | ⊢ ℲxF |
nffo.2 | ⊢ ℲxA |
nffo.3 | ⊢ ℲxB |
Ref | Expression |
---|---|
nffo | ⊢ Ⅎx F:A–onto→B |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fo 4793 | . 2 ⊢ (F:A–onto→B ↔ (F Fn A ∧ ran F = B)) | |
2 | nffo.1 | . . . 4 ⊢ ℲxF | |
3 | nffo.2 | . . . 4 ⊢ ℲxA | |
4 | 2, 3 | nffn 5180 | . . 3 ⊢ Ⅎx F Fn A |
5 | 2 | nfrn 4963 | . . . 4 ⊢ Ⅎxran F |
6 | nffo.3 | . . . 4 ⊢ ℲxB | |
7 | 5, 6 | nfeq 2496 | . . 3 ⊢ Ⅎxran F = B |
8 | 4, 7 | nfan 1824 | . 2 ⊢ Ⅎx(F Fn A ∧ ran F = B) |
9 | 1, 8 | nfxfr 1570 | 1 ⊢ Ⅎx F:A–onto→B |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 Ⅎwnf 1544 = wceq 1642 Ⅎwnfc 2476 ran crn 4773 Fn wfn 4776 –onto→wfo 4779 |
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-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-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 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-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-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-fo 4793 |
This theorem is referenced by: nff1o 5285 |
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