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Theorem f1omptsnlem 35059
 Description: This is the core of the proof of f1omptsn 35060, 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 2758 . . . . . . 7 {𝑥} = {𝑥}
3 snex 5303 . . . . . . . 8 {𝑥} ∈ V
4 eqsbc3 3744 . . . . . . . 8 ({𝑥} ∈ V → ([{𝑥} / 𝑢]𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
53, 4ax-mp 5 . . . . . . 7 ([{𝑥} / 𝑢]𝑢 = {𝑥} ↔ {𝑥} = {𝑥})
62, 5mpbir 234 . . . . . 6 [{𝑥} / 𝑢]𝑢 = {𝑥}
7 sbcel2 4315 . . . . . . . 8 ([{𝑥} / 𝑢]𝑥𝐴𝑥{𝑥} / 𝑢𝐴)
8 csbconstg 3826 . . . . . . . . . 10 ({𝑥} ∈ V → {𝑥} / 𝑢𝐴 = 𝐴)
93, 8ax-mp 5 . . . . . . . . 9 {𝑥} / 𝑢𝐴 = 𝐴
109eleq2i 2843 . . . . . . . 8 (𝑥{𝑥} / 𝑢𝐴𝑥𝐴)
117, 10bitri 278 . . . . . . 7 ([{𝑥} / 𝑢]𝑥𝐴𝑥𝐴)
12 f1omptsn.r . . . . . . . . . . . . . 14 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
1312abeq2i 2887 . . . . . . . . . . . . 13 (𝑢𝑅 ↔ ∃𝑥𝐴 𝑢 = {𝑥})
14 df-rex 3076 . . . . . . . . . . . . 13 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥𝐴𝑢 = {𝑥}))
1513, 14sylbbr 239 . . . . . . . . . . . 12 (∃𝑥(𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
161519.23bi 2188 . . . . . . . . . . 11 ((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
1716sbcth 3713 . . . . . . . . . 10 ({𝑥} ∈ V → [{𝑥} / 𝑢]((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅))
183, 17ax-mp 5 . . . . . . . . 9 [{𝑥} / 𝑢]((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
19 sbcimg 3746 . . . . . . . . . 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 233 . . . . . . . 8 ([{𝑥} / 𝑢](𝑥𝐴𝑢 = {𝑥}) → [{𝑥} / 𝑢]𝑢𝑅)
22 sbcan 3747 . . . . . . . 8 ([{𝑥} / 𝑢](𝑥𝐴𝑢 = {𝑥}) ↔ ([{𝑥} / 𝑢]𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}))
23 sbcel1v 3765 . . . . . . . 8 ([{𝑥} / 𝑢]𝑢𝑅 ↔ {𝑥} ∈ 𝑅)
2421, 22, 233imtr3i 294 . . . . . . 7 (([{𝑥} / 𝑢]𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
2511, 24sylanbr 585 . . . . . 6 ((𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
266, 25mpan2 690 . . . . 5 (𝑥𝐴 → {𝑥} ∈ 𝑅)
271, 26fmpti 6872 . . . 4 𝐹:𝐴𝑅
281fvmpt2 6774 . . . . . . . . 9 ((𝑥𝐴 ∧ {𝑥} ∈ 𝑅) → (𝐹𝑥) = {𝑥})
2926, 28mpdan 686 . . . . . . . 8 (𝑥𝐴 → (𝐹𝑥) = {𝑥})
30 sneq 4535 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
3130, 1, 3fvmpt3i 6768 . . . . . . . 8 (𝑦𝐴 → (𝐹𝑦) = {𝑦})
3229, 31eqeqan12d 2775 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) ↔ {𝑥} = {𝑦}))
33 vex 3413 . . . . . . . 8 𝑥 ∈ V
34 sneqbg 4734 . . . . . . . 8 (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
3533, 34ax-mp 5 . . . . . . 7 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
3632, 35bitrdi 290 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝑥 = 𝑦))
3736biimpd 232 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3837rgen2 3132 . . . 4 𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
39 dff13 7010 . . . 4 (𝐹:𝐴1-1𝑅 ↔ (𝐹:𝐴𝑅 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
4027, 38, 39mpbir2an 710 . . 3 𝐹:𝐴1-1𝑅
41 f1f1orn 6617 . . 3 (𝐹:𝐴1-1𝑅𝐹:𝐴1-1-onto→ran 𝐹)
4240, 41ax-mp 5 . 2 𝐹:𝐴1-1-onto→ran 𝐹
43 rnmptsn 35058 . . . 4 ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
441rneqi 5782 . . . 4 ran 𝐹 = ran (𝑥𝐴 ↦ {𝑥})
4543, 44, 123eqtr4i 2791 . . 3 ran 𝐹 = 𝑅
46 f1oeq3 6596 . . 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 233 1 𝐹:𝐴1-1-onto𝑅
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071  Vcvv 3409  [wsbc 3698  ⦋csb 3807  {csn 4525   ↦ cmpt 5115  ran crn 5528  ⟶wf 6335  –1-1→wf1 6336  –1-1-onto→wf1o 6338  ‘cfv 6339 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347 This theorem is referenced by:  f1omptsn  35060
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