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Theorem f1od 5726
 Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
Hypotheses
Ref Expression
f1od.1 F = (x A C)
f1od.2 ((φ x A) → C W)
f1od.3 ((φ y B) → D X)
f1od.4 (φ → ((x A y = C) ↔ (y B x = D)))
Assertion
Ref Expression
f1od (φF:A1-1-ontoB)
Distinct variable groups:   x,y,A   x,B,y   y,C   x,D   φ,x,y
Allowed substitution hints:   C(x)   D(y)   F(x,y)   W(x,y)   X(x,y)

Proof of Theorem f1od
StepHypRef Expression
1 f1od.2 . . . 4 ((φ x A) → C W)
21ralrimiva 2697 . . 3 (φx A C W)
3 f1od.1 . . . 4 F = (x A C)
43fnmpt 5689 . . 3 (x A C WF Fn A)
52, 4syl 15 . 2 (φF Fn A)
6 f1od.3 . . . . 5 ((φ y B) → D X)
76ralrimiva 2697 . . . 4 (φy B D X)
8 eqid 2353 . . . . 5 (y B D) = (y B D)
98fnmpt 5689 . . . 4 (y B D X → (y B D) Fn B)
107, 9syl 15 . . 3 (φ → (y B D) Fn B)
11 f1od.4 . . . . . 6 (φ → ((x A y = C) ↔ (y B x = D)))
1211opabbidv 4625 . . . . 5 (φ → {y, x (x A y = C)} = {y, x (y B x = D)})
13 df-mpt 5652 . . . . . . . 8 (x A C) = {x, y (x A y = C)}
143, 13eqtri 2373 . . . . . . 7 F = {x, y (x A y = C)}
1514cnveqi 4887 . . . . . 6 F = {x, y (x A y = C)}
16 cnvopab 5030 . . . . . 6 {x, y (x A y = C)} = {y, x (x A y = C)}
1715, 16eqtri 2373 . . . . 5 F = {y, x (x A y = C)}
18 df-mpt 5652 . . . . 5 (y B D) = {y, x (y B x = D)}
1912, 17, 183eqtr4g 2410 . . . 4 (φF = (y B D))
2019fneq1d 5175 . . 3 (φ → (F Fn B ↔ (y B D) Fn B))
2110, 20mpbird 223 . 2 (φF Fn B)
22 dff1o4 5294 . 2 (F:A1-1-ontoB ↔ (F Fn A F Fn B))
235, 21, 22sylanbrc 645 1 (φF:A1-1-ontoB)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  {copab 4622  ◡ccnv 4771   Fn wfn 4776  –1-1-onto→wf1o 4780   ↦ cmpt 5651 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-mpt 5652 This theorem is referenced by:  f1o2d  5727
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