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Theorem f1omptsnlem 32812
Description: This is the core of the proof of f1omptsn 32813, 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 = (𝑥𝐴 ↦ {𝑥})
f1omptsn.r 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
Assertion
Ref Expression
f1omptsnlem 𝐹:𝐴1-1-onto𝑅
Distinct variable groups:   𝑥,𝐴,𝑢   𝑥,𝐹   𝑢,𝑅,𝑥
Allowed substitution hint:   𝐹(𝑢)

Proof of Theorem f1omptsnlem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 f1omptsn.f . . . . 5 𝐹 = (𝑥𝐴 ↦ {𝑥})
2 eqid 2621 . . . . . . 7 {𝑥} = {𝑥}
3 snex 4869 . . . . . . . 8 {𝑥} ∈ V
4 eqsbc3 3457 . . . . . . . 8 ({𝑥} ∈ V → ([{𝑥} / 𝑢]𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
53, 4ax-mp 5 . . . . . . 7 ([{𝑥} / 𝑢]𝑢 = {𝑥} ↔ {𝑥} = {𝑥})
62, 5mpbir 221 . . . . . 6 [{𝑥} / 𝑢]𝑢 = {𝑥}
7 sbcel2 3961 . . . . . . . 8 ([{𝑥} / 𝑢]𝑥𝐴𝑥{𝑥} / 𝑢𝐴)
8 csbconstg 3527 . . . . . . . . . 10 ({𝑥} ∈ V → {𝑥} / 𝑢𝐴 = 𝐴)
93, 8ax-mp 5 . . . . . . . . 9 {𝑥} / 𝑢𝐴 = 𝐴
109eleq2i 2690 . . . . . . . 8 (𝑥{𝑥} / 𝑢𝐴𝑥𝐴)
117, 10bitri 264 . . . . . . 7 ([{𝑥} / 𝑢]𝑥𝐴𝑥𝐴)
12 f1omptsn.r . . . . . . . . . . . . . 14 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
1312abeq2i 2732 . . . . . . . . . . . . 13 (𝑢𝑅 ↔ ∃𝑥𝐴 𝑢 = {𝑥})
14 df-rex 2913 . . . . . . . . . . . . 13 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥𝐴𝑢 = {𝑥}))
1513, 14sylbbr 226 . . . . . . . . . . . 12 (∃𝑥(𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
161519.23bi 2059 . . . . . . . . . . 11 ((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
1716sbcth 3432 . . . . . . . . . 10 ({𝑥} ∈ V → [{𝑥} / 𝑢]((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅))
183, 17ax-mp 5 . . . . . . . . 9 [{𝑥} / 𝑢]((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
19 sbcimg 3459 . . . . . . . . . 10 ({𝑥} ∈ V → ([{𝑥} / 𝑢]((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅) ↔ ([{𝑥} / 𝑢](𝑥𝐴𝑢 = {𝑥}) → [{𝑥} / 𝑢]𝑢𝑅)))
203, 19ax-mp 5 . . . . . . . . 9 ([{𝑥} / 𝑢]((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅) ↔ ([{𝑥} / 𝑢](𝑥𝐴𝑢 = {𝑥}) → [{𝑥} / 𝑢]𝑢𝑅))
2118, 20mpbi 220 . . . . . . . 8 ([{𝑥} / 𝑢](𝑥𝐴𝑢 = {𝑥}) → [{𝑥} / 𝑢]𝑢𝑅)
22 sbcan 3460 . . . . . . . 8 ([{𝑥} / 𝑢](𝑥𝐴𝑢 = {𝑥}) ↔ ([{𝑥} / 𝑢]𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}))
23 sbcel1v 3477 . . . . . . . 8 ([{𝑥} / 𝑢]𝑢𝑅 ↔ {𝑥} ∈ 𝑅)
2421, 22, 233imtr3i 280 . . . . . . 7 (([{𝑥} / 𝑢]𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
2511, 24sylanbr 490 . . . . . 6 ((𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
266, 25mpan2 706 . . . . 5 (𝑥𝐴 → {𝑥} ∈ 𝑅)
271, 26fmpti 6339 . . . 4 𝐹:𝐴𝑅
281fvmpt2 6248 . . . . . . . . 9 ((𝑥𝐴 ∧ {𝑥} ∈ 𝑅) → (𝐹𝑥) = {𝑥})
2926, 28mpdan 701 . . . . . . . 8 (𝑥𝐴 → (𝐹𝑥) = {𝑥})
30 sneq 4158 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
3130, 1, 3fvmpt3i 6244 . . . . . . . 8 (𝑦𝐴 → (𝐹𝑦) = {𝑦})
3229, 31eqeqan12d 2637 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) ↔ {𝑥} = {𝑦}))
33 vex 3189 . . . . . . . 8 𝑥 ∈ V
34 sneqbg 4342 . . . . . . . 8 (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
3533, 34ax-mp 5 . . . . . . 7 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
3632, 35syl6bb 276 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝑥 = 𝑦))
3736biimpd 219 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3837rgen2a 2971 . . . 4 𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
39 dff13 6466 . . . 4 (𝐹:𝐴1-1𝑅 ↔ (𝐹:𝐴𝑅 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
4027, 38, 39mpbir2an 954 . . 3 𝐹:𝐴1-1𝑅
41 f1f1orn 6105 . . 3 (𝐹:𝐴1-1𝑅𝐹:𝐴1-1-onto→ran 𝐹)
4240, 41ax-mp 5 . 2 𝐹:𝐴1-1-onto→ran 𝐹
43 rnmptsn 32811 . . . 4 ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
441rneqi 5312 . . . 4 ran 𝐹 = ran (𝑥𝐴 ↦ {𝑥})
4543, 44, 123eqtr4i 2653 . . 3 ran 𝐹 = 𝑅
46 f1oeq3 6086 . . 3 (ran 𝐹 = 𝑅 → (𝐹:𝐴1-1-onto→ran 𝐹𝐹:𝐴1-1-onto𝑅))
4745, 46ax-mp 5 . 2 (𝐹:𝐴1-1-onto→ran 𝐹𝐹:𝐴1-1-onto𝑅)
4842, 47mpbi 220 1 𝐹:𝐴1-1-onto𝑅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3186  [wsbc 3417  csb 3514  {csn 4148  cmpt 4673  ran crn 5075  wf 5843  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by:  f1omptsn  32813
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