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Theorem ofoaid1 41258
Description: Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
Assertion
Ref Expression
ofoaid1 (((𝐴𝑉𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵m 𝐴)) → (𝐹f +o (𝐴 × {∅})) = 𝐹)

Proof of Theorem ofoaid1
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 simpll 765 . 2 (((𝐴𝑉𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵m 𝐴)) → 𝐴𝑉)
2 onss 7666 . . . . . . 7 (𝐵 ∈ On → 𝐵 ⊆ On)
3 sstr 3934 . . . . . . . 8 ((ran 𝐹𝐵𝐵 ⊆ On) → ran 𝐹 ⊆ On)
43expcom 415 . . . . . . 7 (𝐵 ⊆ On → (ran 𝐹𝐵 → ran 𝐹 ⊆ On))
52, 4syl 17 . . . . . 6 (𝐵 ∈ On → (ran 𝐹𝐵 → ran 𝐹 ⊆ On))
65anim2d 613 . . . . 5 (𝐵 ∈ On → ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ On)))
7 df-f 6462 . . . . 5 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
8 df-f 6462 . . . . 5 (𝐹:𝐴⟶On ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ On))
96, 7, 83imtr4g 296 . . . 4 (𝐵 ∈ On → (𝐹:𝐴𝐵𝐹:𝐴⟶On))
10 elmapi 8668 . . . 4 (𝐹 ∈ (𝐵m 𝐴) → 𝐹:𝐴𝐵)
119, 10impel 507 . . 3 ((𝐵 ∈ On ∧ 𝐹 ∈ (𝐵m 𝐴)) → 𝐹:𝐴⟶On)
1211adantll 712 . 2 (((𝐴𝑉𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵m 𝐴)) → 𝐹:𝐴⟶On)
13 peano1 7767 . . 3 ∅ ∈ ω
14 fnconstg 6692 . . 3 (∅ ∈ ω → (𝐴 × {∅}) Fn 𝐴)
1513, 14mp1i 13 . 2 (((𝐴𝑉𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵m 𝐴)) → (𝐴 × {∅}) Fn 𝐴)
16 simp2 1137 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐹:𝐴⟶On)
1716ffnd 6631 . . . 4 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐹 Fn 𝐴)
18 simp3 1138 . . . 4 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐴 × {∅}) Fn 𝐴)
19 simp1 1136 . . . 4 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → 𝐴𝑉)
20 inidm 4158 . . . 4 (𝐴𝐴) = 𝐴
2117, 18, 19, 19, 20offn 7578 . . 3 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐹f +o (𝐴 × {∅})) Fn 𝐴)
2217, 18jca 513 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐹 Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
2322adantr 482 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → (𝐹 Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴))
2419adantr 482 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → 𝐴𝑉)
25 simpr 486 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → 𝑎𝐴)
26 fnfvof 7582 . . . . 5 (((𝐹 Fn 𝐴 ∧ (𝐴 × {∅}) Fn 𝐴) ∧ (𝐴𝑉𝑎𝐴)) → ((𝐹f +o (𝐴 × {∅}))‘𝑎) = ((𝐹𝑎) +o ((𝐴 × {∅})‘𝑎)))
2723, 24, 25, 26syl12anc 835 . . . 4 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → ((𝐹f +o (𝐴 × {∅}))‘𝑎) = ((𝐹𝑎) +o ((𝐴 × {∅})‘𝑎)))
28 fvconst2g 7109 . . . . . 6 ((∅ ∈ ω ∧ 𝑎𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
2913, 25, 28sylancr 588 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → ((𝐴 × {∅})‘𝑎) = ∅)
3029oveq2d 7323 . . . 4 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → ((𝐹𝑎) +o ((𝐴 × {∅})‘𝑎)) = ((𝐹𝑎) +o ∅))
3116ffvelcdmda 6993 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → (𝐹𝑎) ∈ On)
32 oa0 8377 . . . . 5 ((𝐹𝑎) ∈ On → ((𝐹𝑎) +o ∅) = (𝐹𝑎))
3331, 32syl 17 . . . 4 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → ((𝐹𝑎) +o ∅) = (𝐹𝑎))
3427, 30, 333eqtrd 2780 . . 3 (((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) ∧ 𝑎𝐴) → ((𝐹f +o (𝐴 × {∅}))‘𝑎) = (𝐹𝑎))
3521, 17, 34eqfnfvd 6944 . 2 ((𝐴𝑉𝐹:𝐴⟶On ∧ (𝐴 × {∅}) Fn 𝐴) → (𝐹f +o (𝐴 × {∅})) = 𝐹)
361, 12, 15, 35syl3anc 1371 1 (((𝐴𝑉𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵m 𝐴)) → (𝐹f +o (𝐴 × {∅})) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1087   = wceq 1539  wcel 2104  wss 3892  c0 4262  {csn 4565   × cxp 5598  ran crn 5601  Oncon0 6281   Fn wfn 6453  wf 6454  cfv 6458  (class class class)co 7307  f cof 7563  ωcom 7744   +o coa 8325  m cmap 8646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3340  df-rab 3341  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-of 7565  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-oadd 8332  df-map 8648
This theorem is referenced by: (None)
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