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Theorem isf32lem1 9490
Description: Lemma for isfin3-2 9504. 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 6433 . . . . 5 (𝑎 = 𝐵 → (𝐹𝑎) = (𝐹𝐵))
21sseq1d 3857 . . . 4 (𝑎 = 𝐵 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹𝐵) ⊆ (𝐹𝐵)))
32imbi2d 332 . . 3 (𝑎 = 𝐵 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹𝐵) ⊆ (𝐹𝐵))))
4 fveq2 6433 . . . . 5 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
54sseq1d 3857 . . . 4 (𝑎 = 𝑏 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹𝑏) ⊆ (𝐹𝐵)))
65imbi2d 332 . . 3 (𝑎 = 𝑏 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹𝑏) ⊆ (𝐹𝐵))))
7 fveq2 6433 . . . . 5 (𝑎 = suc 𝑏 → (𝐹𝑎) = (𝐹‘suc 𝑏))
87sseq1d 3857 . . . 4 (𝑎 = suc 𝑏 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹𝐵)))
98imbi2d 332 . . 3 (𝑎 = suc 𝑏 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹𝐵))))
10 fveq2 6433 . . . . 5 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
1110sseq1d 3857 . . . 4 (𝑎 = 𝐴 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹𝐴) ⊆ (𝐹𝐵)))
1211imbi2d 332 . . 3 (𝑎 = 𝐴 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹𝐴) ⊆ (𝐹𝐵))))
13 ssid 3848 . . . 4 (𝐹𝐵) ⊆ (𝐹𝐵)
14132a1i 12 . . 3 (𝐵 ∈ ω → (𝜑 → (𝐹𝐵) ⊆ (𝐹𝐵)))
15 isf32lem.b . . . . . . 7 (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
16 suceq 6028 . . . . . . . . . 10 (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏)
1716fveq2d 6437 . . . . . . . . 9 (𝑥 = 𝑏 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑏))
18 fveq2 6433 . . . . . . . . 9 (𝑥 = 𝑏 → (𝐹𝑥) = (𝐹𝑏))
1917, 18sseq12d 3859 . . . . . . . 8 (𝑥 = 𝑏 → ((𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹𝑏)))
2019rspcv 3522 . . . . . . 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 717 . . . . 5 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑏) → (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹𝑏)))
23 sstr2 3834 . . . . 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 7354 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝜑 → (𝐹𝐴) ⊆ (𝐹𝐵)))
2726impr 448 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → (𝐹𝐴) ⊆ (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1656  wcel 2164  wral 3117  wss 3798  𝒫 cpw 4378   cint 4697  ran crn 5343  suc csuc 5965  wf 6119  cfv 6123  ωcom 7326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-tr 4976  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fv 6131  df-om 7327
This theorem is referenced by:  isf32lem2  9491  isf32lem3  9492
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