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Theorem pwfilem 8437
Description: Lemma for pwfi 8438. (Contributed by NM, 26-Mar-2007.)
Hypothesis
Ref Expression
pwfilem.1 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))
Assertion
Ref Expression
pwfilem (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
Distinct variable groups:   𝑏,𝑐   𝑥,𝑐
Allowed substitution hints:   𝐹(𝑥,𝑏,𝑐)

Proof of Theorem pwfilem
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 pwundif 5168 . 2 𝒫 (𝑏 ∪ {𝑥}) = ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏)
2 vex 3358 . . . . . . . . 9 𝑐 ∈ V
3 snex 5050 . . . . . . . . 9 {𝑥} ∈ V
42, 3unex 7124 . . . . . . . 8 (𝑐 ∪ {𝑥}) ∈ V
5 pwfilem.1 . . . . . . . 8 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))
64, 5fnmpti 6173 . . . . . . 7 𝐹 Fn 𝒫 𝑏
7 dffn4 6277 . . . . . . 7 (𝐹 Fn 𝒫 𝑏𝐹:𝒫 𝑏onto→ran 𝐹)
86, 7mpbi 221 . . . . . 6 𝐹:𝒫 𝑏onto→ran 𝐹
9 fodomfi 8416 . . . . . 6 ((𝒫 𝑏 ∈ Fin ∧ 𝐹:𝒫 𝑏onto→ran 𝐹) → ran 𝐹 ≼ 𝒫 𝑏)
108, 9mpan2 672 . . . . 5 (𝒫 𝑏 ∈ Fin → ran 𝐹 ≼ 𝒫 𝑏)
11 domfi 8358 . . . . 5 ((𝒫 𝑏 ∈ Fin ∧ ran 𝐹 ≼ 𝒫 𝑏) → ran 𝐹 ∈ Fin)
1210, 11mpdan 668 . . . 4 (𝒫 𝑏 ∈ Fin → ran 𝐹 ∈ Fin)
13 eldifi 3890 . . . . . . . . 9 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}))
143elpwun 7145 . . . . . . . . 9 (𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}) ↔ (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
1513, 14sylib 209 . . . . . . . 8 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏)
16 undif1 4195 . . . . . . . . 9 ((𝑑 ∖ {𝑥}) ∪ {𝑥}) = (𝑑 ∪ {𝑥})
17 elpwunsn 4373 . . . . . . . . . . 11 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑥𝑑)
1817snssd 4486 . . . . . . . . . 10 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → {𝑥} ⊆ 𝑑)
19 ssequn2 3944 . . . . . . . . . 10 ({𝑥} ⊆ 𝑑 ↔ (𝑑 ∪ {𝑥}) = 𝑑)
2018, 19sylib 209 . . . . . . . . 9 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∪ {𝑥}) = 𝑑)
2116, 20syl5req 2821 . . . . . . . 8 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
22 uneq1 3918 . . . . . . . . . 10 (𝑐 = (𝑑 ∖ {𝑥}) → (𝑐 ∪ {𝑥}) = ((𝑑 ∖ {𝑥}) ∪ {𝑥}))
2322eqeq2d 2784 . . . . . . . . 9 (𝑐 = (𝑑 ∖ {𝑥}) → (𝑑 = (𝑐 ∪ {𝑥}) ↔ 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})))
2423rspcev 3465 . . . . . . . 8 (((𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
2515, 21, 24syl2anc 574 . . . . . . 7 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
265, 4elrnmpti 5526 . . . . . . 7 (𝑑 ∈ ran 𝐹 ↔ ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥}))
2725, 26sylibr 225 . . . . . 6 (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ ran 𝐹)
2827ssriv 3762 . . . . 5 (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹
29 ssdomg 8176 . . . . 5 (ran 𝐹 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹 → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹))
3012, 28, 29mpisyl 21 . . . 4 (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹)
31 domfi 8358 . . . 4 ((ran 𝐹 ∈ Fin ∧ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹) → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
3212, 30, 31syl2anc 574 . . 3 (𝒫 𝑏 ∈ Fin → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin)
33 unfi 8404 . . 3 (((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin ∧ 𝒫 𝑏 ∈ Fin) → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
3432, 33mpancom 669 . 2 (𝒫 𝑏 ∈ Fin → ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) ∈ Fin)
351, 34syl5eqel 2857 1 (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1634  wcel 2148  wrex 3065  cdif 3726  cun 3727  wss 3729  𝒫 cpw 4307  {csn 4326   class class class wbr 4797  cmpt 4876  ran crn 5264   Fn wfn 6037  ontowfo 6040  cdom 8128  Fincfn 8130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-int 4623  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-1o 7734  df-oadd 7738  df-er 7917  df-en 8131  df-dom 8132  df-fin 8134
This theorem is referenced by:  pwfi  8438
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