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Theorem f1omptsnlem 33495
Description: This is the core of the proof of f1omptsn 33496, 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 2806 . . . . . . 7 {𝑥} = {𝑥}
3 snex 5098 . . . . . . . 8 {𝑥} ∈ V
4 eqsbc3 3673 . . . . . . . 8 ({𝑥} ∈ V → ([{𝑥} / 𝑢]𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
53, 4ax-mp 5 . . . . . . 7 ([{𝑥} / 𝑢]𝑢 = {𝑥} ↔ {𝑥} = {𝑥})
62, 5mpbir 222 . . . . . 6 [{𝑥} / 𝑢]𝑢 = {𝑥}
7 sbcel2 4186 . . . . . . . 8 ([{𝑥} / 𝑢]𝑥𝐴𝑥{𝑥} / 𝑢𝐴)
8 csbconstg 3741 . . . . . . . . . 10 ({𝑥} ∈ V → {𝑥} / 𝑢𝐴 = 𝐴)
93, 8ax-mp 5 . . . . . . . . 9 {𝑥} / 𝑢𝐴 = 𝐴
109eleq2i 2877 . . . . . . . 8 (𝑥{𝑥} / 𝑢𝐴𝑥𝐴)
117, 10bitri 266 . . . . . . 7 ([{𝑥} / 𝑢]𝑥𝐴𝑥𝐴)
12 f1omptsn.r . . . . . . . . . . . . . 14 𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
1312abeq2i 2919 . . . . . . . . . . . . 13 (𝑢𝑅 ↔ ∃𝑥𝐴 𝑢 = {𝑥})
14 df-rex 3102 . . . . . . . . . . . . 13 (∃𝑥𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥𝐴𝑢 = {𝑥}))
1513, 14sylbbr 227 . . . . . . . . . . . 12 (∃𝑥(𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
161519.23bi 2226 . . . . . . . . . . 11 ((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
1716sbcth 3648 . . . . . . . . . 10 ({𝑥} ∈ V → [{𝑥} / 𝑢]((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅))
183, 17ax-mp 5 . . . . . . . . 9 [{𝑥} / 𝑢]((𝑥𝐴𝑢 = {𝑥}) → 𝑢𝑅)
19 sbcimg 3675 . . . . . . . . . 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 221 . . . . . . . 8 ([{𝑥} / 𝑢](𝑥𝐴𝑢 = {𝑥}) → [{𝑥} / 𝑢]𝑢𝑅)
22 sbcan 3676 . . . . . . . 8 ([{𝑥} / 𝑢](𝑥𝐴𝑢 = {𝑥}) ↔ ([{𝑥} / 𝑢]𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}))
23 sbcel1v 3692 . . . . . . . 8 ([{𝑥} / 𝑢]𝑢𝑅 ↔ {𝑥} ∈ 𝑅)
2421, 22, 233imtr3i 282 . . . . . . 7 (([{𝑥} / 𝑢]𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
2511, 24sylanbr 573 . . . . . 6 ((𝑥𝐴[{𝑥} / 𝑢]𝑢 = {𝑥}) → {𝑥} ∈ 𝑅)
266, 25mpan2 674 . . . . 5 (𝑥𝐴 → {𝑥} ∈ 𝑅)
271, 26fmpti 6600 . . . 4 𝐹:𝐴𝑅
281fvmpt2 6508 . . . . . . . . 9 ((𝑥𝐴 ∧ {𝑥} ∈ 𝑅) → (𝐹𝑥) = {𝑥})
2926, 28mpdan 670 . . . . . . . 8 (𝑥𝐴 → (𝐹𝑥) = {𝑥})
30 sneq 4380 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
3130, 1, 3fvmpt3i 6504 . . . . . . . 8 (𝑦𝐴 → (𝐹𝑦) = {𝑦})
3229, 31eqeqan12d 2822 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) ↔ {𝑥} = {𝑦}))
33 vex 3394 . . . . . . . 8 𝑥 ∈ V
34 sneqbg 4562 . . . . . . . 8 (𝑥 ∈ V → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
3533, 34ax-mp 5 . . . . . . 7 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
3632, 35syl6bb 278 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝑥 = 𝑦))
3736biimpd 220 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3837rgen2a 3165 . . . 4 𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)
39 dff13 6732 . . . 4 (𝐹:𝐴1-1𝑅 ↔ (𝐹:𝐴𝑅 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
4027, 38, 39mpbir2an 693 . . 3 𝐹:𝐴1-1𝑅
41 f1f1orn 6360 . . 3 (𝐹:𝐴1-1𝑅𝐹:𝐴1-1-onto→ran 𝐹)
4240, 41ax-mp 5 . 2 𝐹:𝐴1-1-onto→ran 𝐹
43 rnmptsn 33494 . . . 4 ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}
441rneqi 5553 . . . 4 ran 𝐹 = ran (𝑥𝐴 ↦ {𝑥})
4543, 44, 123eqtr4i 2838 . . 3 ran 𝐹 = 𝑅
46 f1oeq3 6341 . . 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 221 1 𝐹:𝐴1-1-onto𝑅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wex 1859  wcel 2156  {cab 2792  wral 3096  wrex 3097  Vcvv 3391  [wsbc 3633  csb 3728  {csn 4370  cmpt 4923  ran crn 5312  wf 6093  1-1wf1 6094  1-1-ontowf1o 6096  cfv 6097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105
This theorem is referenced by:  f1omptsn  33496
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