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Mirrors > Home > ILE Home > Th. List > f1oen2g | Unicode version |
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6644 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
f1oen2g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1of 5360 | . . . 4 | |
2 | fex2 5286 | . . . 4 | |
3 | 1, 2 | syl3an1 1249 | . . 3 |
4 | 3 | 3coml 1188 | . 2 |
5 | simp3 983 | . 2 | |
6 | f1oen3g 6641 | . 2 | |
7 | 4, 5, 6 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 962 wcel 1480 cvv 2681 class class class wbr 3924 wf 5114 wf1o 5117 cen 6625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-en 6628 |
This theorem is referenced by: f1oeng 6644 enrefg 6651 en2d 6655 en3d 6656 ener 6666 f1imaen2g 6680 cnven 6695 xpcomen 6714 xpfi 6811 nnenom 10200 |
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