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Theorem f1omptsnlem 36309
Description: This is the core of the proof of f1omptsn 36310, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 15-Jul-2020.)
Hypotheses
Ref Expression
f1omptsn.f 𝐹 = (π‘₯ ∈ 𝐴 ↦ {π‘₯})
f1omptsn.r 𝑅 = {𝑒 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑒 = {π‘₯}}
Assertion
Ref Expression
f1omptsnlem 𝐹:𝐴–1-1-onto→𝑅
Distinct variable groups:   π‘₯,𝐴,𝑒   π‘₯,𝐹   𝑒,𝑅,π‘₯
Allowed substitution hint:   𝐹(𝑒)

Proof of Theorem f1omptsnlem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 f1omptsn.f . . . . 5 𝐹 = (π‘₯ ∈ 𝐴 ↦ {π‘₯})
2 eqid 2732 . . . . . . 7 {π‘₯} = {π‘₯}
3 vsnex 5429 . . . . . . . 8 {π‘₯} ∈ V
4 eqsbc1 3826 . . . . . . . 8 ({π‘₯} ∈ V β†’ ([{π‘₯} / 𝑒]𝑒 = {π‘₯} ↔ {π‘₯} = {π‘₯}))
53, 4ax-mp 5 . . . . . . 7 ([{π‘₯} / 𝑒]𝑒 = {π‘₯} ↔ {π‘₯} = {π‘₯})
62, 5mpbir 230 . . . . . 6 [{π‘₯} / 𝑒]𝑒 = {π‘₯}
7 sbcel2 4415 . . . . . . . 8 ([{π‘₯} / 𝑒]π‘₯ ∈ 𝐴 ↔ π‘₯ ∈ ⦋{π‘₯} / π‘’β¦Œπ΄)
8 csbconstg 3912 . . . . . . . . . 10 ({π‘₯} ∈ V β†’ ⦋{π‘₯} / π‘’β¦Œπ΄ = 𝐴)
93, 8ax-mp 5 . . . . . . . . 9 ⦋{π‘₯} / π‘’β¦Œπ΄ = 𝐴
109eleq2i 2825 . . . . . . . 8 (π‘₯ ∈ ⦋{π‘₯} / π‘’β¦Œπ΄ ↔ π‘₯ ∈ 𝐴)
117, 10bitri 274 . . . . . . 7 ([{π‘₯} / 𝑒]π‘₯ ∈ 𝐴 ↔ π‘₯ ∈ 𝐴)
12 f1omptsn.r . . . . . . . . . . . . . 14 𝑅 = {𝑒 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑒 = {π‘₯}}
1312eqabri 2877 . . . . . . . . . . . . 13 (𝑒 ∈ 𝑅 ↔ βˆƒπ‘₯ ∈ 𝐴 𝑒 = {π‘₯})
14 df-rex 3071 . . . . . . . . . . . . 13 (βˆƒπ‘₯ ∈ 𝐴 𝑒 = {π‘₯} ↔ βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ 𝑒 = {π‘₯}))
1513, 14sylbbr 235 . . . . . . . . . . . 12 (βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ 𝑒 = {π‘₯}) β†’ 𝑒 ∈ 𝑅)
161519.23bi 2184 . . . . . . . . . . 11 ((π‘₯ ∈ 𝐴 ∧ 𝑒 = {π‘₯}) β†’ 𝑒 ∈ 𝑅)
1716sbcth 3792 . . . . . . . . . 10 ({π‘₯} ∈ V β†’ [{π‘₯} / 𝑒]((π‘₯ ∈ 𝐴 ∧ 𝑒 = {π‘₯}) β†’ 𝑒 ∈ 𝑅))
183, 17ax-mp 5 . . . . . . . . 9 [{π‘₯} / 𝑒]((π‘₯ ∈ 𝐴 ∧ 𝑒 = {π‘₯}) β†’ 𝑒 ∈ 𝑅)
19 sbcimg 3828 . . . . . . . . . 10 ({π‘₯} ∈ V β†’ ([{π‘₯} / 𝑒]((π‘₯ ∈ 𝐴 ∧ 𝑒 = {π‘₯}) β†’ 𝑒 ∈ 𝑅) ↔ ([{π‘₯} / 𝑒](π‘₯ ∈ 𝐴 ∧ 𝑒 = {π‘₯}) β†’ [{π‘₯} / 𝑒]𝑒 ∈ 𝑅)))
203, 19ax-mp 5 . . . . . . . . 9 ([{π‘₯} / 𝑒]((π‘₯ ∈ 𝐴 ∧ 𝑒 = {π‘₯}) β†’ 𝑒 ∈ 𝑅) ↔ ([{π‘₯} / 𝑒](π‘₯ ∈ 𝐴 ∧ 𝑒 = {π‘₯}) β†’ [{π‘₯} / 𝑒]𝑒 ∈ 𝑅))
2118, 20mpbi 229 . . . . . . . 8 ([{π‘₯} / 𝑒](π‘₯ ∈ 𝐴 ∧ 𝑒 = {π‘₯}) β†’ [{π‘₯} / 𝑒]𝑒 ∈ 𝑅)
22 sbcan 3829 . . . . . . . 8 ([{π‘₯} / 𝑒](π‘₯ ∈ 𝐴 ∧ 𝑒 = {π‘₯}) ↔ ([{π‘₯} / 𝑒]π‘₯ ∈ 𝐴 ∧ [{π‘₯} / 𝑒]𝑒 = {π‘₯}))
23 sbcel1v 3848 . . . . . . . 8 ([{π‘₯} / 𝑒]𝑒 ∈ 𝑅 ↔ {π‘₯} ∈ 𝑅)
2421, 22, 233imtr3i 290 . . . . . . 7 (([{π‘₯} / 𝑒]π‘₯ ∈ 𝐴 ∧ [{π‘₯} / 𝑒]𝑒 = {π‘₯}) β†’ {π‘₯} ∈ 𝑅)
2511, 24sylanbr 582 . . . . . 6 ((π‘₯ ∈ 𝐴 ∧ [{π‘₯} / 𝑒]𝑒 = {π‘₯}) β†’ {π‘₯} ∈ 𝑅)
266, 25mpan2 689 . . . . 5 (π‘₯ ∈ 𝐴 β†’ {π‘₯} ∈ 𝑅)
271, 26fmpti 7113 . . . 4 𝐹:π΄βŸΆπ‘…
281fvmpt2 7009 . . . . . . . . 9 ((π‘₯ ∈ 𝐴 ∧ {π‘₯} ∈ 𝑅) β†’ (πΉβ€˜π‘₯) = {π‘₯})
2926, 28mpdan 685 . . . . . . . 8 (π‘₯ ∈ 𝐴 β†’ (πΉβ€˜π‘₯) = {π‘₯})
30 sneq 4638 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ {π‘₯} = {𝑦})
3130, 1, 3fvmpt3i 7003 . . . . . . . 8 (𝑦 ∈ 𝐴 β†’ (πΉβ€˜π‘¦) = {𝑦})
3229, 31eqeqan12d 2746 . . . . . . 7 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ↔ {π‘₯} = {𝑦}))
33 vex 3478 . . . . . . . 8 π‘₯ ∈ V
34 sneqbg 4844 . . . . . . . 8 (π‘₯ ∈ V β†’ ({π‘₯} = {𝑦} ↔ π‘₯ = 𝑦))
3533, 34ax-mp 5 . . . . . . 7 ({π‘₯} = {𝑦} ↔ π‘₯ = 𝑦)
3632, 35bitrdi 286 . . . . . 6 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ↔ π‘₯ = 𝑦))
3736biimpd 228 . . . . 5 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) β†’ π‘₯ = 𝑦))
3837rgen2 3197 . . . 4 βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) β†’ π‘₯ = 𝑦)
39 dff13 7256 . . . 4 (𝐹:𝐴–1-1→𝑅 ↔ (𝐹:π΄βŸΆπ‘… ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) β†’ π‘₯ = 𝑦)))
4027, 38, 39mpbir2an 709 . . 3 𝐹:𝐴–1-1→𝑅
41 f1f1orn 6844 . . 3 (𝐹:𝐴–1-1→𝑅 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
4240, 41ax-mp 5 . 2 𝐹:𝐴–1-1-ontoβ†’ran 𝐹
43 rnmptsn 36308 . . . 4 ran (π‘₯ ∈ 𝐴 ↦ {π‘₯}) = {𝑒 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑒 = {π‘₯}}
441rneqi 5936 . . . 4 ran 𝐹 = ran (π‘₯ ∈ 𝐴 ↦ {π‘₯})
4543, 44, 123eqtr4i 2770 . . 3 ran 𝐹 = 𝑅
46 f1oeq3 6823 . . 3 (ran 𝐹 = 𝑅 β†’ (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ 𝐹:𝐴–1-1-onto→𝑅))
4745, 46ax-mp 5 . 2 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ↔ 𝐹:𝐴–1-1-onto→𝑅)
4842, 47mpbi 229 1 𝐹:𝐴–1-1-onto→𝑅
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474  [wsbc 3777  β¦‹csb 3893  {csn 4628   ↦ cmpt 5231  ran crn 5677  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551
This theorem is referenced by:  f1omptsn  36310
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