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Theorem f1omptsnlem 33494
Description: This is the core of the proof of f1omptsn 33495, 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 2760 . . . . . . 7 {𝑥} = {𝑥}
3 snex 5057 . . . . . . . 8 {𝑥} ∈ V
4 eqsbc3 3616 . . . . . . . 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 4132 . . . . . . . 8 ([{𝑥} / 𝑢]𝑥𝐴𝑥{𝑥} / 𝑢𝐴)
8 csbconstg 3687 . . . . . . . . . 10 ({𝑥} ∈ V → {𝑥} / 𝑢𝐴 = 𝐴)
93, 8ax-mp 5 . . . . . . . . 9 {𝑥} / 𝑢𝐴 = 𝐴
109eleq2i 2831 . . . . . . . 8 (𝑥{𝑥} / 𝑢𝐴𝑥𝐴)
117, 10bitri 264 . . . . . . 7 ([{𝑥} / 𝑢]𝑥𝐴𝑥𝐴)
12 f1omptsn.r . . . . . . . . . . . . . 14 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
1312abeq2i 2873 . . . . . . . . . . . . 13 (𝑢𝑅 ↔ ∃𝑥𝐴 𝑢 = {𝑥})
14 df-rex 3056 . . . . . . . . . . . . 13 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥𝐴𝑢 = {𝑥}))
1513, 14sylbbr 226 . . . . . . . . . . . 12 (∃𝑥(𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
161519.23bi 2208 . . . . . . . . . . 11 ((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
1716sbcth 3591 . . . . . . . . . 10 ({𝑥} ∈ V → [{𝑥} / 𝑢]((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅))
183, 17ax-mp 5 . . . . . . . . 9 [{𝑥} / 𝑢]((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
19 sbcimg 3618 . . . . . . . . . 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 3619 . . . . . . . 8 ([{𝑥} / 𝑢](𝑥𝐴𝑢 = {𝑥}) ↔ ([{𝑥} / 𝑢]𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}))
23 sbcel1v 3636 . . . . . . . 8 ([{𝑥} / 𝑢]𝑢𝑅 ↔ {𝑥} ∈ 𝑅)
2421, 22, 233imtr3i 280 . . . . . . 7 (([{𝑥} / 𝑢]𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
2511, 24sylanbr 491 . . . . . 6 ((𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
266, 25mpan2 709 . . . . 5 (𝑥𝐴 → {𝑥} ∈ 𝑅)
271, 26fmpti 6546 . . . 4 𝐹:𝐴𝑅
281fvmpt2 6453 . . . . . . . . 9 ((𝑥𝐴 ∧ {𝑥} ∈ 𝑅) → (𝐹𝑥) = {𝑥})
2926, 28mpdan 705 . . . . . . . 8 (𝑥𝐴 → (𝐹𝑥) = {𝑥})
30 sneq 4331 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
3130, 1, 3fvmpt3i 6449 . . . . . . . 8 (𝑦𝐴 → (𝐹𝑦) = {𝑦})
3229, 31eqeqan12d 2776 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) ↔ {𝑥} = {𝑦}))
33 vex 3343 . . . . . . . 8 𝑥 ∈ V
34 sneqbg 4519 . . . . . . . 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 3115 . . . 4 𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
39 dff13 6675 . . . 4 (𝐹:𝐴1-1𝑅 ↔ (𝐹:𝐴𝑅 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
4027, 38, 39mpbir2an 993 . . 3 𝐹:𝐴1-1𝑅
41 f1f1orn 6309 . . 3 (𝐹:𝐴1-1𝑅𝐹:𝐴1-1-onto→ran 𝐹)
4240, 41ax-mp 5 . 2 𝐹:𝐴1-1-onto→ran 𝐹
43 rnmptsn 33493 . . . 4 ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
441rneqi 5507 . . . 4 ran 𝐹 = ran (𝑥𝐴 ↦ {𝑥})
4543, 44, 123eqtr4i 2792 . . 3 ran 𝐹 = 𝑅
46 f1oeq3 6290 . . 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 383   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wral 3050  wrex 3051  Vcvv 3340  [wsbc 3576  csb 3674  {csn 4321  cmpt 4881  ran crn 5267  wf 6045  1-1wf1 6046  1-1-ontowf1o 6048  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057
This theorem is referenced by:  f1omptsn  33495
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