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Theorem isf32lem1 9764
Description: Lemma for isfin3-2 9778. 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 6652 . . . . 5 (𝑎 = 𝐵 → (𝐹𝑎) = (𝐹𝐵))
21sseq1d 3973 . . . 4 (𝑎 = 𝐵 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹𝐵) ⊆ (𝐹𝐵)))
32imbi2d 344 . . 3 (𝑎 = 𝐵 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹𝐵) ⊆ (𝐹𝐵))))
4 fveq2 6652 . . . . 5 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
54sseq1d 3973 . . . 4 (𝑎 = 𝑏 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹𝑏) ⊆ (𝐹𝐵)))
65imbi2d 344 . . 3 (𝑎 = 𝑏 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹𝑏) ⊆ (𝐹𝐵))))
7 fveq2 6652 . . . . 5 (𝑎 = suc 𝑏 → (𝐹𝑎) = (𝐹‘suc 𝑏))
87sseq1d 3973 . . . 4 (𝑎 = suc 𝑏 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹𝐵)))
98imbi2d 344 . . 3 (𝑎 = suc 𝑏 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹𝐵))))
10 fveq2 6652 . . . . 5 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
1110sseq1d 3973 . . . 4 (𝑎 = 𝐴 → ((𝐹𝑎) ⊆ (𝐹𝐵) ↔ (𝐹𝐴) ⊆ (𝐹𝐵)))
1211imbi2d 344 . . 3 (𝑎 = 𝐴 → ((𝜑 → (𝐹𝑎) ⊆ (𝐹𝐵)) ↔ (𝜑 → (𝐹𝐴) ⊆ (𝐹𝐵))))
13 ssid 3964 . . . 4 (𝐹𝐵) ⊆ (𝐹𝐵)
14132a1i 12 . . 3 (𝐵 ∈ ω → (𝜑 → (𝐹𝐵) ⊆ (𝐹𝐵)))
15 isf32lem.b . . . . . . 7 (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
16 suceq 6234 . . . . . . . . . 10 (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏)
1716fveq2d 6656 . . . . . . . . 9 (𝑥 = 𝑏 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑏))
18 fveq2 6652 . . . . . . . . 9 (𝑥 = 𝑏 → (𝐹𝑥) = (𝐹𝑏))
1917, 18sseq12d 3975 . . . . . . . 8 (𝑥 = 𝑏 → ((𝐹‘suc 𝑥) ⊆ (𝐹𝑥) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹𝑏)))
2019rspcv 3593 . . . . . . 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 725 . . . . 5 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑏) → (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹𝑏)))
23 sstr2 3949 . . . . 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 7595 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝜑 → (𝐹𝐴) ⊆ (𝐹𝐵)))
2726impr 458 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → (𝐹𝐴) ⊆ (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  wss 3908  𝒫 cpw 4511   cint 4851  ran crn 5533  suc csuc 6171  wf 6330  cfv 6334  ωcom 7565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fv 6342  df-om 7566
This theorem is referenced by:  isf32lem2  9765  isf32lem3  9766
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