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Theorem isf32lem1 10387
Description: Lemma for isfin3-2 10401. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.)
Hypotheses
Ref Expression
isf32lem.a (𝜑𝐹:ω⟶𝒫 𝐺)
isf32lem.b (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
isf32lem.c (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)
Assertion
Ref Expression
isf32lem1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → (𝐹𝐴) ⊆ (𝐹𝐵))
Distinct variable groups:   𝑥,𝐵   𝜑,𝑥   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem isf32lem1
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6893 . . . . 5 (𝑎 = 𝐵 → (𝐹𝑎) = (𝐹𝐵))
21sseq1d 4010 . . . 4 (𝑎 = 𝐵 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹𝐵) ⊆ (𝐹𝐵)))
32imbi2d 339 . . 3 (𝑎 = 𝐵 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹𝐵) ⊆ (𝐹𝐵))))
4 fveq2 6893 . . . . 5 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
54sseq1d 4010 . . . 4 (𝑎 = 𝑏 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹𝑏) ⊆ (𝐹𝐵)))
65imbi2d 339 . . 3 (𝑎 = 𝑏 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹𝑏) ⊆ (𝐹𝐵))))
7 fveq2 6893 . . . . 5 (𝑎 = suc 𝑏 → (𝐹𝑎) = (𝐹‘suc 𝑏))
87sseq1d 4010 . . . 4 (𝑎 = suc 𝑏 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹𝐵)))
98imbi2d 339 . . 3 (𝑎 = suc 𝑏 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹𝐵))))
10 fveq2 6893 . . . . 5 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
1110sseq1d 4010 . . . 4 (𝑎 = 𝐴 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹𝐴) ⊆ (𝐹𝐵)))
1211imbi2d 339 . . 3 (𝑎 = 𝐴 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹𝐴) ⊆ (𝐹𝐵))))
13 ssid 4001 . . . 4 (𝐹𝐵) ⊆ (𝐹𝐵)
14132a1i 12 . . 3 (𝐵 ∈ ω → (𝜑 → (𝐹𝐵) ⊆ (𝐹𝐵)))
15 isf32lem.b . . . . . . 7 (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
16 suceq 6434 . . . . . . . . . 10 (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏)
1716fveq2d 6897 . . . . . . . . 9 (𝑥 = 𝑏 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑏))
18 fveq2 6893 . . . . . . . . 9 (𝑥 = 𝑏 → (𝐹𝑥) = (𝐹𝑏))
1917, 18sseq12d 4012 . . . . . . . 8 (𝑥 = 𝑏 → ((𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹𝑏)))
2019rspcv 3603 . . . . . . 7 (𝑏 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥) → (𝐹‘suc 𝑏) ⊆ (𝐹𝑏)))
2115, 20syl5 34 . . . . . 6 (𝑏 ∈ ω → (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹𝑏)))
2221ad2antrr 724 . . . . 5 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑏) → (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹𝑏)))
23 sstr2 3985 . . . . 5 ((𝐹‘suc 𝑏) ⊆ (𝐹𝑏) → ((𝐹𝑏) ⊆ (𝐹𝐵) → (𝐹‘suc 𝑏) ⊆ (𝐹𝐵)))
2422, 23syl6 35 . . . 4 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑏) → (𝜑 → ((𝐹𝑏) ⊆ (𝐹𝐵) → (𝐹‘suc 𝑏) ⊆ (𝐹𝐵))))
2524a2d 29 . . 3 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑏) → ((𝜑 → (𝐹𝑏) ⊆ (𝐹𝐵)) → (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹𝐵))))
263, 6, 9, 12, 14, 25findsg 7902 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝜑 → (𝐹𝐴) ⊆ (𝐹𝐵)))
2726impr 453 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → (𝐹𝐴) ⊆ (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1534  wcel 2099  wral 3051  wss 3946  𝒫 cpw 4597   cint 4946  ran crn 5675  suc csuc 6370  wf 6542  cfv 6546  ωcom 7868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-tr 5263  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fv 6554  df-om 7869
This theorem is referenced by:  isf32lem2  10388  isf32lem3  10389
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