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Mirrors > Home > MPE Home > Th. List > dff12 | Structured version Visualization version GIF version |
Description: Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.) |
Ref | Expression |
---|---|
dff12 | ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1 6187 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
2 | funcnv2 6249 | . . 3 ⊢ (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 𝑥𝐹𝑦) | |
3 | 2 | anbi2i 613 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦)) |
4 | 1, 3 | bitri 267 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 ∀wal 1505 ∃*wmo 2542 class class class wbr 4923 ◡ccnv 5399 Fun wfun 6176 ⟶wf 6178 –1-1→wf1 6179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rab 3091 df-v 3411 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4924 df-opab 4986 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-fun 6184 df-f1 6187 |
This theorem is referenced by: dff13 6832 fseqenlem2 9237 s4f1o 14132 2ndcdisj 21758 usgrexmplef 26734 phpreu 34265 |
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