Theorem weisoeq2 6749

Description: Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 7301. (Contributed by Mario Carneiro, 25-Jun-2015.)

Assertion

Ref	Expression
weisoeq2	⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)

Proof of Theorem weisoeq2

Step	Hyp	Ref	Expression
1		isocnv 6723	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴))
2		isocnv 6723	. . . 4 ⊢ (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐺 Isom 𝑆, 𝑅 (𝐵, 𝐴))
3	1, 2	anim12i 600	. . 3 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → (◡𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴) ∧ ◡𝐺 Isom 𝑆, 𝑅 (𝐵, 𝐴)))
4		weisoeq 6748	. . 3 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (◡𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴) ∧ ◡𝐺 Isom 𝑆, 𝑅 (𝐵, 𝐴))) → ◡𝐹 = ◡𝐺)
5	3, 4	sylan2 580	. 2 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → ◡𝐹 = ◡𝐺)
6		simprl 754	. . . 4 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
7		isof1o 6716	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
8		f1orel 6281	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹)
9	6, 7, 8	3syl 18	. . 3 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → Rel 𝐹)
10		simprr 756	. . . 4 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))
11		isof1o 6716	. . . 4 ⊢ (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐺:𝐴–1-1-onto→𝐵)
12		f1orel 6281	. . . 4 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → Rel 𝐺)
13	10, 11, 12	3syl 18	. . 3 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → Rel 𝐺)
14		cnveqb 5731	. . 3 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → (𝐹 = 𝐺 ↔ ◡𝐹 = ◡𝐺))
15	9, 13, 14	syl2anc 573	. 2 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (𝐹 = 𝐺 ↔ ◡𝐹 = ◡𝐺))
16	5, 15	mpbird 247	1 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   Se wse 5206   We wwe 5207  ^◡ccnv 5248  Rel wrel 5254  –1-1-onto→wf1o 6030   Isom wiso 6032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040

This theorem is referenced by:  wemoiso2  7301