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Theorem f1omptsnlem 37319
Description: This is the core of the proof of f1omptsn 37320, 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 2735 . . . . . . 7 {𝑥} = {𝑥}
3 vsnex 5440 . . . . . . . 8 {𝑥} ∈ V
4 eqsbc1 3841 . . . . . . . 8 ({𝑥} ∈ V → ([{𝑥} / 𝑢]𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
53, 4ax-mp 5 . . . . . . 7 ([{𝑥} / 𝑢]𝑢 = {𝑥} ↔ {𝑥} = {𝑥})
62, 5mpbir 231 . . . . . 6 [{𝑥} / 𝑢]𝑢 = {𝑥}
7 sbcel2 4424 . . . . . . . 8 ([{𝑥} / 𝑢]𝑥𝐴𝑥{𝑥} / 𝑢𝐴)
8 csbconstg 3927 . . . . . . . . . 10 ({𝑥} ∈ V → {𝑥} / 𝑢𝐴 = 𝐴)
93, 8ax-mp 5 . . . . . . . . 9 {𝑥} / 𝑢𝐴 = 𝐴
109eleq2i 2831 . . . . . . . 8 (𝑥{𝑥} / 𝑢𝐴𝑥𝐴)
117, 10bitri 275 . . . . . . 7 ([{𝑥} / 𝑢]𝑥𝐴𝑥𝐴)
12 f1omptsn.r . . . . . . . . . . . . . 14 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
1312eqabri 2883 . . . . . . . . . . . . 13 (𝑢𝑅 ↔ ∃𝑥𝐴 𝑢 = {𝑥})
14 df-rex 3069 . . . . . . . . . . . . 13 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥𝐴𝑢 = {𝑥}))
1513, 14sylbbr 236 . . . . . . . . . . . 12 (∃𝑥(𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
161519.23bi 2189 . . . . . . . . . . 11 ((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
1716sbcth 3806 . . . . . . . . . 10 ({𝑥} ∈ V → [{𝑥} / 𝑢]((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅))
183, 17ax-mp 5 . . . . . . . . 9 [{𝑥} / 𝑢]((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
19 sbcimg 3843 . . . . . . . . . 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 230 . . . . . . . 8 ([{𝑥} / 𝑢](𝑥𝐴𝑢 = {𝑥}) → [{𝑥} / 𝑢]𝑢𝑅)
22 sbcan 3844 . . . . . . . 8 ([{𝑥} / 𝑢](𝑥𝐴𝑢 = {𝑥}) ↔ ([{𝑥} / 𝑢]𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}))
23 sbcel1v 3862 . . . . . . . 8 ([{𝑥} / 𝑢]𝑢𝑅 ↔ {𝑥} ∈ 𝑅)
2421, 22, 233imtr3i 291 . . . . . . 7 (([{𝑥} / 𝑢]𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
2511, 24sylanbr 582 . . . . . 6 ((𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
266, 25mpan2 691 . . . . 5 (𝑥𝐴 → {𝑥} ∈ 𝑅)
271, 26fmpti 7132 . . . 4 𝐹:𝐴𝑅
281fvmpt2 7027 . . . . . . . . 9 ((𝑥𝐴 ∧ {𝑥} ∈ 𝑅) → (𝐹𝑥) = {𝑥})
2926, 28mpdan 687 . . . . . . . 8 (𝑥𝐴 → (𝐹𝑥) = {𝑥})
30 sneq 4641 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
3130, 1, 3fvmpt3i 7021 . . . . . . . 8 (𝑦𝐴 → (𝐹𝑦) = {𝑦})
3229, 31eqeqan12d 2749 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) ↔ {𝑥} = {𝑦}))
33 vex 3482 . . . . . . . 8 𝑥 ∈ V
34 sneqbg 4848 . . . . . . . 8 (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
3533, 34ax-mp 5 . . . . . . 7 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
3632, 35bitrdi 287 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝑥 = 𝑦))
3736biimpd 229 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3837rgen2 3197 . . . 4 𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
39 dff13 7275 . . . 4 (𝐹:𝐴1-1𝑅 ↔ (𝐹:𝐴𝑅 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
4027, 38, 39mpbir2an 711 . . 3 𝐹:𝐴1-1𝑅
41 f1f1orn 6860 . . 3 (𝐹:𝐴1-1𝑅𝐹:𝐴1-1-onto→ran 𝐹)
4240, 41ax-mp 5 . 2 𝐹:𝐴1-1-onto→ran 𝐹
43 rnmptsn 37318 . . . 4 ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
441rneqi 5951 . . . 4 ran 𝐹 = ran (𝑥𝐴 ↦ {𝑥})
4543, 44, 123eqtr4i 2773 . . 3 ran 𝐹 = 𝑅
46 f1oeq3 6839 . . 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 230 1 𝐹:𝐴1-1-onto𝑅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  [wsbc 3791  csb 3908  {csn 4631  cmpt 5231  ran crn 5690  wf 6559  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by:  f1omptsn  37320
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